TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II:
DECOMPOSITION
GROUPS
AND
ENDOMORPHISMS
Shinichi
Mochizuki
October
2013
Abstract.
The
present
paper,
which
forms
the
second
part
of
a
three-part
series
in
which
we
study
absolute
anabelian
geometry
from
an
algorithmic
point
of
view,
focuses
on
the
study
of
the
closely
related
notions
of
decomposition
groups
and
endomorphisms
in
this
anabelian
context.
We
begin
by
studying
an
abstract
combinatorial
analogue
of
the
algebro-geometric
notion
of
a
stable
polycurve
[i.e.,
a
“successive
extension
of
families
of
stable
curves”]
and
showing
that
the
“geometry
of
log
divisors
on
stable
polycurves”
may
be
extended,
in
a
purely
group-theoretic
fashion,
to
this
abstract
combinatorial
analogue;
this
leads
to
various
anabelian
results
concerning
configuration
spaces.
We
then
turn
to
the
study
of
the
absolute
pro-Σ
anabelian
geometry
of
hyperbolic
curves
over
mixed-characteristic
local
fields,
for
Σ
a
set
of
primes
of
cardinality
≥
2
that
contains
the
residue
characteristic
of
the
base
field.
In
particular,
we
prove
a
certain
“pro-p
resolution
of
nonsingularities”
type
result,
which
implies
a
“conditional”
anabelian
result
to
the
effect
that
the
condition,
on
an
isomorphism
of
arithmetic
fundamental
groups,
of
preservation
of
decomposition
groups
of
“most”
closed
points
implies
that
the
isomorphism
arises
from
an
isomorphism
of
schemes
—
i.e.,
in
a
word,
“point-theoreticity
implies
geometricity”;
a
“non-conditional”
version
of
this
result
is
then
obtained
for
“pro-
curves”
obtained
by
removing
from
a
proper
curve
some
set
of
closed
points
which
is
“p-adically
dense
in
a
Galois-compatible
fashion”.
Finally,
we
study,
from
an
algorithmic
point
of
view,
the
theory
of
Belyi
and
elliptic
cuspidalizations,
i.e.,
group-theoretic
reconstruction
algorithms
for
the
arithmetic
fundamental
group
of
an
open
subscheme
of
a
hyperbolic
curve
that
arise
from
consideration
of
certain
endomorphisms
determined
by
Belyi
maps
and
endomorphisms
of
elliptic
curves.
Contents:
§0.
Notations
and
Conventions
§1.
A
Combinatorial
Analogue
of
Stable
Polycurves
§2.
Geometric
Uniformly
Toral
Neighborhoods
§3.
Elliptic
and
Belyi
Cuspidalizations
Introduction
In
the
present
paper,
which
forms
the
second
part
of
a
three-part
series,
we
continue
our
discussion
of
various
topics
in
absolute
anabelian
geometry
from
2000
Mathematical
Subject
Classification.
Primary
14H30;
Secondary
14H25.
Typeset
by
AMS-TEX
1
2
SHINICHI
MOCHIZUKI
a
“group-theoretic
algorithmic”
point
of
view,
as
discussed
in
the
Introduction
to
[Mzk15].
The
topics
presented
in
the
present
paper
center
around
the
following
two
themes:
(A)
[the
subgroups
of
arithmetic
fundamental
groups
constituted
by]
de-
composition
groups
of
subvarieties
of
a
given
variety
[such
as
closed
points,
divisors]
as
a
crucial
tool
that
leads
to
absolute
anabelian
results;
(B)
“hidden
endomorphisms”
—
which
may
be
thought
of
as
“hidden
symmetries”
—
of
hyperbolic
curves
that
give
rise
to
various
absolute
anabelian
results.
In
fact,
“decomposition
groups”
and
“endomorphisms”
are,
in
a
certain
sense,
re-
lated
notions
—
that
is
to
say,
the
monoid
of
“endomorphisms”
of
a
variety
may
be
thought
of
as
a
sort
of
“decomposition
group
of
the
generic
point”!
With
regard
to
the
theme
(B),
we
recall
that
the
endomorphisms
of
an
abelian
variety
play
a
fundamental
role
in
the
theory
of
abelian
varieties
[e.g.,
elliptic
curves!].
Unlike
abelian
varieties,
hyperbolic
curves
[say,
in
characteristic
zero]
do
not
have
sufficient
“endomorphisms”
in
the
literal,
scheme-theoretic
sense
to
form
the
basis
for
an
interesting
theory.
This
difference
between
abelian
varieties
and
hyperbolic
curves
may
be
thought
of,
at
a
certain
level,
as
reflecting
the
dif-
ference
between
linear
Euclidean
geometries
and
non-linear
hyperbolic
geometries.
From
this
point
of
view,
it
is
natural
to
search
for
“hidden
endomorphisms”
that
are,
in
some
way,
related
to
the
intrinsic
non-linear
hyperbolic
geometry
of
a
hy-
perbolic
curve.
Examples
[that
appear
in
previous
papers
of
the
author]
of
such
“hidden
endomorphisms”
—
which
exhibit
a
remarkable
tendency
to
be
related
[for
instance,
via
some
induced
action
on
the
arithmetic
fundamental
group]
to
some
sort
of
“anabelian
result”
—
are
the
following:
(i)
the
interpretation
of
the
automorphism
group
P
SL
2
(R)
of
the
uni-
versal
covering
of
a
hyperbolic
Riemann
surface
as
an
object
that
gives
rise
to
a
certain
“Grothendieck
Conjecture-type
result”
in
the
“geometry
of
categories”
[cf.
[Mzk11],
Theorem
1.12];
(ii)
the
interpretation
of
the
theory
of
Teichmüller
mappings
[a
sort
of
endomorphism
—
cf.
(iii)
below]
between
hyperbolic
Riemann
surfaces
as
a
“Grothendieck
Conjecture-type
result”
in
the
“geometry
of
categories”
[cf.
[Mzk11],
Theorem
2.3];
(iii)
the
use
of
the
endomorphisms
constituted
by
Frobenius
liftings
—
in
the
form
of
p-adic
Teichmüller
theory
—
to
obtain
the
absolute
anabelian
result
constituted
by
[Mzk6],
Corollary
3.8;
(iv)
the
use
of
the
endomorphism
rings
of
Lubin-Tate
groups
to
obtain
the
absolute
anabelian
result
constituted
by
[Mzk15],
Corollaries
3.8,
3.9.
The
main
results
of
the
present
paper
—
in
which
both
themes
(A)
and
(B)
play
a
central
role
—
are
the
following:
(1)
In
§1,
we
develop
a
purely
combinatorial
approach
to
the
algebro-geometric
notion
of
a
stable
polycurve
[cf.
[Mzk2],
Definition
4.5].
This
approach
may
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
be
thought
of
as
being
motivated
by
the
purely
combinatorial
approach
to
the
notion
of
a
stable
curve
given
in
[Mzk13].
Moreover,
in
§1,
we
apply
the
theory
of
[Mzk13]
to
give,
in
effect,
“group-theoretic
algorithms”
for
reconstructing
the
“abstract
combinatorial
analogue”
of
the
geome-
try
of
the
various
divisors
—
in
the
form
of
inertia
and
decomposition
groups
—
associated
to
the
canonical
log
structure
of
a
stable
polycurve
[cf.
Theorem
1.7].
These
techniques,
together
with
the
theory
of
[MT],
give
rise
to
various
relative
and
absolute
anabelian
results
concerning
configuration
spaces
associated
to
hyperbolic
curves
[cf.
Corollaries
1.10,
1.11].
Relative
to
the
discussion
above
of
“hidden
en-
domorphisms”,
we
observe
that
such
configuration
spaces
may
be
thought
of
as
representing
a
sort
of
“tautological
endomorphism/correspondence”
of
the
hyperbolic
curve
in
question.
(2)
In
§2,
we
study
the
absolute
pro-Σ
anabelian
geometry
of
hyperbolic
curves
over
mixed-characteristic
local
fields,
for
Σ
a
set
of
primes
of
car-
dinality
≥
2
that
contains
the
residue
characteristic
of
the
base
field.
In
particular,
we
show
that
the
condition,
on
an
isomorphism
of
arithmetic
fundamental
groups,
of
preservation
of
decomposition
groups
of
“most”
closed
points
implies
that
the
isomorphism
arises
from
an
isomorphism
of
schemes
—
i.e.,
in
a
word,
“point-theoreticity
implies
geometricity”
[cf.
Corollary
2.9].
This
condition
may
be
removed
if
one
works
with
“pro-
curves”
obtained
by
removing
from
a
proper
curve
some
set
of
closed
points
which
is
“p-adically
dense
in
a
Galois-compatible
fashion”
[cf.
Corollary
2.10].
The
key
technical
result
that
underlies
these
anabelian
results
is
a
certain
“pro-p
resolution
of
nonsingularities”
type
result
[cf.
Lemma
2.6;
Remark
2.6.1;
Corollary
2.11]
—
i.e.,
a
result
reminiscent
of
the
main
[profinite]
results
of
[Tama2].
This
technical
result
allows
one
to
apply
the
theory
of
uniformly
toral
neighborhoods
developed
in
[Mzk15],
§3.
Relative
to
the
discussion
above
of
“hidden
endomorphisms”,
this
technical
result
is
interesting
[cf.,
e.g.,
(iii)
above]
in
that
one
central
step
of
the
proof
of
the
technical
result
is
quite
similar
to
the
well-known
classical
argument
that
implies
the
nonexistence
of
a
Frobenius
lifting
for
stable
curves
over
the
ring
of
Witt
vectors
of
a
finite
field
[cf.
Remark
2.6.2].
(3)
In
§3,
we
re-examine
the
theory
of
[Mzk8],
§2,
for
reconstructing
the
decomposition
groups
of
closed
points
from
the
point
of
view
of
the
present
series
of
developing
“group-theoretic
algorithms”.
In
particular,
we
observe
that
these
group-theoretic
algorithms
allow
one
to
use
Belyi
maps
and
endomorphisms
of
elliptic
curves
to
con-
struct
[not
only
decomposition
groups
of
closed
points,
but
also]
“cuspidalizations”
3
4
SHINICHI
MOCHIZUKI
[i.e.,
the
full
arithmetic
fundamental
groups
of
the
open
subschemes
ob-
tained
by
removing
various
closed
points
—
cf.
the
theory
of
[Mzk14]]
as-
sociated
to
various
types
of
closed
points
[cf.
Corollaries
3.3,
3.4,
3.7,
3.8].
Relative
to
the
discussion
above
of
“hidden
endomorphisms”,
the
theory
of
Belyi
and
elliptic
cuspidalizations
given
in
§3
illustrates
quite
explicitly
how
endomorphisms
[arising
from
Belyi
maps
or
endomorphisms
of
elliptic
curves]
can
give
rise
to
group-theoretic
reconstruction
algorithms.
Finally,
we
remark
that
although
the
“algorithmic
approach”
to
stating
anabelian
results
is
not
carried
out
very
explicitly
in
§1,
§2
[by
comparison
to
§3
or
[Mzk15]],
the
translation
into
“algorithmic
language”
of
the
more
traditional
“Grothendieck
Conjecture-type”
statements
of
the
main
results
of
§1,
§2
is
quite
routine.
[Here,
it
should
be
noted
that
the
results
of
§1
that
depend
on
“Uchida’s
theorem”
—
i.e.,
Theorem
1.8,
(ii);
Corollary
1.11,
(iv)
—
constitute
a
notatable
exception
to
this
“remark”,
an
exception
that
will
be
discussed
in
more
detail
in
[Mzk16]
—
cf.,
e.g.,
[Mzk16],
Remark
1.9.5.]
That
is
to
say,
this
translation
was
not
carried
out
explicitly
by
the
author
solely
because
of
the
complexity
of
the
algorithms
implicit
in
§1,
§2,
i.e.,
not
as
a
result
of
any
substantive
mathematical
obstacles.
Acknowledgements:
I
would
like
to
thank
Akio
Tamagawa
and
Yuichiro
Hoshi
for
many
helpful
discussions
concerning
the
material
presented
in
this
paper
and
Emmanuel
Lepage
for
a
suggestion
that
led
to
Remark
2.11.1,
(i).
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
5
Section
0:
Notations
and
Conventions
We
shall
continue
to
use
the
“Notations
and
Conventions”
of
[Mzk15],
§0.
In
addition,
we
shall
use
the
following
notation
and
conventions:
Topological
Groups:
Let
G
be
a
topologically
finitely
generated,
slim
profinite
group.
Thus,
G
admits
a
basis
of
characteristic
open
subgroups.
Any
such
basis
determines
a
profinite
topology
on
the
groups
Aut(G),
Out(G).
If
ρ
:
H
→
Out(G)
is
any
continuous
homomorphism
of
profinite
groups,
then
we
denote
by
out
G
H
the
profinite
group
obtained
by
pulling
back
the
natural
exact
sequence
of
profinite
groups
1
→
G
→
Aut(G)
→
Out(G)
→
1
via
ρ.
Thus,
we
have
a
natural
exact
out
sequence
of
profinite
groups
1
→
G
→
G
H
→
H
→
1.
Semi-graphs:
Let
Γ
be
a
connected
semi-graph
[cf.,
e.g.,
[Mzk9],
§1,
for
a
review
of
the
theory
of
semi-graphs].
We
shall
refer
to
the
[possibly
infinite]
dimension
over
Q
of
the
singular
homology
module
H
1
(Γ,
Q)
as
the
loop-rank
lp-rk(Γ)
of
Γ.
We
shall
say
that
Γ
is
loop-ample
if,
for
any
edge
e
of
Γ,
the
semi-graph
obtained
from
Γ
by
removing
e
remains
connected.
We
shall
say
that
Γ
is
untangled
if
every
closed
edge
of
Γ
abuts
to
two
distinct
vertices
[cf.
[Mzk9],
§1].
We
shall
say
that
Γ
is
edge-paired
(respectively,
edge-even)
if
Γ
is
untangled,
and,
moreover,
for
any
two
[not
necessarily
distinct!]
vertices
v,
v
of
Γ,
the
set
of
edges
e
of
Γ
such
that
e
abuts
to
a
vertex
w
of
Γ
if
and
only
if
w
∈
{v,
v
}
is
either
empty
or
of
cardinality
≥
2
(respectively,
empty
or
of
even
cardinality).
[Thus,
one
verifies
immediately
that
if
Γ
is
edge-paired
(respectively,
edge-even),
then
it
is
loop-ample
(respectively,
edge-paired).]
We
shall
refer
to
as
a
simple
path
in
Γ
any
connected
subgraph
γ
⊆
Γ
such
that
the
following
conditions
are
satisfied:
(a)
γ
is
a
finite
tree
that
has
at
least
one
edge;
(b)
given
any
vertex
v
of
γ,
there
exist
at
most
two
branches
of
edges
of
γ
that
abut
to
v.
Thus,
[one
verifies
easily
that]
a
simple
path
γ
has
precisely
two
vertices
v
such
that
there
exists
precisely
one
branch
of
an
edge
of
γ
that
abuts
to
v;
we
shall
refer
to
these
two
vertices
as
the
terminal
vertices
of
the
simple
path
γ.
If
γ,
γ
are
simple
paths
in
Γ
such
that
the
terminal
vertices
of
γ,
γ
coincide,
then
we
shall
say
that
γ,
γ
are
co-terminal.
Log
Schemes:
We
shall
often
regard
a
scheme
as
a
log
scheme
equipped
the
trivial
log
struc-
ture.
Any
fiber
product
of
fs
[i.e.,
fine,
saturated]
log
schemes
is
to
be
taken
in
the
category
of
fs
log
schemes.
In
particular,
the
underlying
scheme
of
such
a
product
is
finite
over,
but
not
necessarily
isomorphic
to,
the
fiber
product
of
the
underlying
schemes.
6
SHINICHI
MOCHIZUKI
Curves:
We
shall
refer
to
a
hyperbolic
orbicurve
X
as
semi-elliptic
[i.e.,
“of
type
(1,
1)
±
”
in
the
terminology
of
[Mzk12],
§0]
if
there
exists
a
finite
étale
double
covering
Y
→
X,
where
Y
is
a
once-punctured
elliptic
curve,
and
the
covering
is
given
by
the
stack-theoretic
quotient
of
Y
by
the
“action
of
±1”
[i.e.,
relative
to
the
group
operation
on
the
elliptic
curve
given
by
the
canonical
compactification
of
Y
].
For
i
=
1,
2,
let
X
i
be
a
hyperbolic
orbicurve
over
a
field
k
i
.
Then
we
shall
say
that
X
1
,
X
2
are
isogenous
[cf.
[Mzk14],
§0]
if
there
exists
a
hyperbolic
orbicurve
X
over
a
field
k,
together
with
finite
étale
morphisms
X
→
X
i
,
for
i
=
1,
2.
Note
that
in
this
situation,
the
morphisms
X
→
X
i
induce
finite
separable
inclusions
of
fields
k
i
→
k.
[Indeed,
this
follows
immediately
from
the
easily
verified
fact
that
×
every
subgroup
G
⊆
Γ(X,
O
X
)
such
that
G
{0}
determines
a
field
is
necessarily
contained
in
k
×
.]
We
shall
use
the
term
stable
log
curve
as
it
was
defined
in
[Mzk9],
§0.
Let
X
log
→
S
log
be
a
stable
log
curve
over
an
fs
log
scheme
S
log
,
where
S
=
Spec(k)
for
some
field
k;
k
a
separable
closure
of
k.
Then
we
shall
refer
to
as
the
loop-rank
lp-rk(X
log
)
[or
lp-rk(X)]
of
X
log
[or
X]
the
loop-rank
of
the
dual
graph
of
X
log
×
k
k
[or
X
×
k
k].
We
shall
say
that
X
log
[or
X]
is
loop-ample
(respectively,
untangled;
edge-paired;
edge-even)
if
the
dual
semi-graph
with
compact
structure
[cf.
[Mzk5],
Appendix]
of
X
log
×
k
k
is
loop-ample
(respectively,
untangled;
edge-paired;
edge-even)
[as
a
connected
semi-graph].
We
shall
say
that
X
log
[or
X]
is
sturdy
if
the
normalization
of
every
irreducible
component
of
X
is
of
genus
≥
2
[cf.
[Mzk13],
Remark
1.1.5].
Observe
that
for
any
prime
number
l
invertible
on
S,
there
exist
an
fs
log
scheme
T
log
over
S
log
,
where
T
=
Spec(k
),
for
some
finite
separable
extension
k
of
k,
and
a
connected
Galois
log
admissible
covering
Y
log
→
X
log
×
S
log
T
log
[cf.
[Mzk1],
§3]
of
degree
a
power
of
l
such
that
Y
log
is
sturdy
and
edge-paired
[hence,
in
particular,
untangled
and
loop-ample];
if,
moreover,
l
=
2,
then
one
may
also
take
Y
log
to
be
edge-even.
[Indeed,
to
verify
this
observation,
we
may
assume
that
k
=
k.
Then
note
that
any
hyperbolic
curve
U
over
k
admits
a
connected
finite
étale
Galois
covering
V
→
U
of
degree
a
power
of
l
such
that
V
is
of
genus
≥
2
and
ramified
with
ramification
index
l
2
at
each
of
the
cusps
of
V
—
cf.
the
discussion
at
the
end
of
the
present
§0.
Thus,
by
gluing
together
such
coverings
at
the
nodes
of
X,
one
concludes
that
there
exists
a
connected
Galois
log
admissible
covering
Z
1
log
→
X
log
×
S
log
T
log
of
degree
a
power
of
l
which
is
totally
ramified
over
every
node
of
X
with
ramification
index
l
2
such
that
every
irreducible
component
of
Z
1
is
of
genus
≥
2
—
i.e.,
Z
1
log
is
sturdy.
Next,
observe
that
there
exists
a
connected
Galois
log
admissible
covering
Z
2
log
→
Z
1
log
of
degree
a
power
of
l
that
arises
from
a
covering
of
the
dual
graph
of
Z
1
such
that
Z
2
log
is
untangled
[and
still
sturdy].
Finally,
observe
that
there
exists
a
connected
Galois
log
admissible
covering
Z
3
log
→
Z
2
log
of
degree
a
positive
power
of
l
which
restricts
to
a
connected
finite
étale
covering
over
every
irreducible
component
of
Z
2
[hence
is
unramified
at
the
nodes]
such
that
Z
3
log
is
edge-paired
[and
still
sturdy
and
untangled]
for
arbitrary
def
l
and
edge-even
when
l
=
2.
Thus,
we
may
take
Y
log
=
Z
3
log
.]
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
7
Observe
that
if
X
is
loop-ample,
then
for
every
point
x
∈
X(k)
which
is
not
a
unique
cusp
of
X
[i.e.,
either
x
is
not
a
cusp
or
if
x
is
a
cusp,
then
it
is
not
the
unique
cusp
of
X],
the
evaluation
map
H
0
(X,
ω
X
log
/S
log
)
→
ω
X
log
/S
log
|
x
is
surjective.
Indeed,
by
considering
the
long
exact
sequence
associated
to
the
short
exact
sequence
0
→
ω
X
log
/S
log
⊗
O
X
I
x
→
ω
X
log
/S
log
→
ω
X
log
/S
log
|
x
→
0,
where
I
x
⊆
O
X
is
the
sheaf
of
ideals
corresponding
to
x,
one
verifies
immediately
that
it
suffices
to
show
that
the
surjection
def
def
H
x
=
H
1
(X,
ω
X
log
/S
log
⊗
O
X
I
x
)
H
=
H
1
(X,
ω
X
log
/S
log
)
is
injective.
If
x
is
a
node,
then
the
fact
that
X
is
loop-ample
implies
[by
computing
via
Serre
duality]
that
either
dim
k
(H
x
)
=
dim
k
(H)
=
1
[if
X
has
no
cusps]
or
dim
k
(H
x
)
=
dim
k
(H)
=
0
[if
X
has
cusps].
Thus,
we
may
assume
that
x
is
not
a
node,
so
the
surjection
H
x
H
is
dual
to
the
injection
def
def
M
=
H
0
(X,
O
X
(−D))
→
M
x
=
H
0
(X,
O
X
(x
−
D))
def
(⊆
N
x
=
H
0
(X,
O
X
(x)))
—
where
we
write
D
⊆
X
for
the
divisor
of
cusps
of
X.
If
x
is
a
cusp,
then
it
follows
that
D
is
of
degree
≥
2,
and
one
computes
easily
that
dim
k
(M)
=
dim
k
(M
x
)
=
0.
Thus,
we
may
assume
that
x
is
not
a
cusp.
Write
C
for
the
irreducible
component
of
X
containing
x.
Now
suppose
that
the
injection
M
→
M
x
is
not
surjective.
Then
it
follows
that
dim
k
(N
x
)
=
2,
and
that
N
x
determines
a
basepoint-free
linear
system
on
X.
In
particular,
N
x
determines
a
morphism
φ
:
X
→
P
1
k
that
is
of
∼
degree
1
on
C
—
i.e.,
determines
an
isomorphism
C
→
P
1
k
—
and
constant
on
the
other
irreducible
components
of
X.
Since
X
is
loop-ample,
it
follows
that
the
dual
graph
Γ
of
X
either
has
no
edges
or
admits
a
loop
containing
the
vertex
determined
by
C.
On
the
other
hand,
the
existence
of
such
a
loop
contradicts
the
fact
that
φ
∼
determines
an
isomorphism
C
→
P
1
k
.
Thus,
we
conclude
that
X
=
C
∼
=
P
1
k
,
so
D
is
of
degree
≥
3.
But
this
implies
that
M
x
=
0,
a
contradiction.
Finally,
let
U
be
a
hyperbolic
curve
over
an
algebraically
closed
field
k
and
l
a
prime
number
invertible
in
k.
Suppose
that
the
cardinality
r
of
the
set
of
cusps
of
U
is
≥
2,
and,
moreover,
that,
if
l
=
2,
then
r
is
even.
Then
observe
that
it
follows
immediately
from
the
well-known
structure
of
the
maximal
pro-l
quotient
of
the
abelianization
of
the
étale
fundamental
group
of
U
that
for
every
power
l
n
of
l,
where
n
is
a
positive
integer,
there
exists
a
cyclic
covering
V
→
U
of
degree
l
n
that
is
totally
ramified
over
the
cusps
of
U
.
Indeed,
this
observation
is
an
immediate
consequence
of
the
elementary
fact
that,
in
light
of
our
assumptions
on
r,
there
always
exist
r
−
1
integers
prime
to
l
whose
sum
is
also
prime
to
l.
We
shall
often
make
use
of
the
assumption
that
a
stable
log
curve
is
edge-paired
—
or,
when
l
=
2,
edge-even
—
by
applying
the
above
observation
to
the
various
connected
components
of
the
complement
of
the
cusps
and
nodes
of
the
stable
log
curve.
8
SHINICHI
MOCHIZUKI
Section
1:
A
Combinatorial
Analogue
of
Stable
Polycurves
In
the
present
§1,
we
apply
the
theory
of
[Mzk13]
to
study
a
sort
of
purely
group-theoretic,
combinatorial
analogue
[cf.
Definition
1.5
below]
of
the
notion
of
a
stable
polycurve
introduced
in
[Mzk2],
Definition
4.5.
This
allows
one
to
reconstruct
the
“abstract
combinatorial
analogue”
of
the
“geometry
of
log
divisors”
[i.e.,
divisors
associated
to
the
log
structure
of
a
stable
polycurve]
of
such
a
combinatorial
object
via
group
theory
[cf.
Theorem
1.7].
Finally,
we
apply
the
theory
of
[MT]
to
obtain
various
consequences
of
the
theory
of
the
present
§1
[cf.
Corollaries
1.10,
1.11]
concerning
the
absolute
anabelian
geometry
of
configuration
spaces.
We
begin
by
recalling
the
discussion
of
[Mzk13],
Example
2.5.
Example
1.1.
Stable
Log
Curves
over
a
Logarithmic
Point
(Revisited).
(i)
Let
k
be
a
separably
closed
field;
Σ
a
nonempty
set
of
prime
numbers
in-
vertible
in
k;
M
⊆
Q
the
monoid
of
positive
rational
numbers
with
denomina-
tors
invertible
in
k;
S
log
(respectively,
T
log
)
the
log
scheme
obtained
by
equipping
def
def
S
=
Spec(k)
(respectively,
T
=
Spec(k))
with
the
log
structure
determined
by
the
chart
N
1
→
0
∈
k
(respectively,
M
1
→
0
∈
k);
T
log
→
S
log
the
morphism
determined
by
the
natural
inclusion
N
→
M;
X
log
→
S
log
def
a
stable
log
curve
over
S
log
.
Thus,
the
profinite
group
I
S
log
=
Aut(T
log
/S
log
)
∼
admits
a
natural
isomorphism
I
S
log
→
Hom(Q/Z,
k
×
)
and
fits
into
an
natural
exact
sequence
def
def
1
→
Δ
X
log
=
π
1
(X
log
×
S
log
T
log
)
→
Π
X
log
=
π
1
(X
log
)
→
I
S
log
→
1
—
where
we
write
“π
1
(−)”
for
the
“log
fundamental
group”
of
the
log
scheme
in
parentheses
[which
amounts,
in
this
case,
to
the
fundamental
group
arising
from
the
admissible
coverings
of
X
log
],
relative
to
an
appropriate
choice
of
basepoint
[cf.
[Ill]
for
a
survey
of
the
theory
of
log
fundamental
groups].
In
particular,
if
we
write
I
S
Σ
log
for
the
maximal
pro-Σ
quotient
of
I
S
log
,
then
as
abstract
profinite
groups,
Σ
,
where
we
write
Z
Σ
for
the
maximal
pro-Σ
quotient
of
Z.
I
S
Σ
log
∼
=
Z
(ii)
On
the
other
hand,
X
log
determines
a
semi-graph
of
anabelioids
[cf.
[Mzk9],
Definition
2.1]
of
pro-Σ
PSC-type
[cf.
[Mzk13],
Definition
1.1,
(i)],
whose
underlying
semi-graph
we
denote
by
G.
Thus,
for
each
vertex
v
[corresponding
to
an
irreducible
component
of
X
log
]
(respectively,
edge
e
[corresponding
to
a
node
or
cusp
of
X
log
])
of
G,
we
have
a
connected
anabelioid
[i.e.,
a
Galois
category]
G
v
(respectively,
G
e
),
and
for
each
branch
b
of
an
edge
e
abutting
to
a
vertex
v,
we
are
given
a
morphism
of
anabelioids
G
e
→
G
v
.
Then
the
maximal
pro-Σ
completion
of
Δ
X
log
may
be
identified
with
the
“PSC-fundamental
group”
Π
G
associated
to
G.
Also,
we
recall
that
Π
G
is
slim
[cf.,
e.g.,
[Mzk13],
Remark
1.1.3],
and
that
the
groups
Aut(G),
Out(Π
G
)
may
be
equipped
with
profinite
topologies
in
such
a
way
that
the
natural
morphism
Aut(G)
→
Out(Π
G
)
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
9
is
a
continuous
injection
[cf.
the
discussion
at
the
beginning
of
[Mzk13],
§2],
which
we
shall
use
to
identify
Aut(G)
with
its
image
in
Out(Π
G
).
In
particular,
we
obtain
a
natural
continuous
homomorphism
I
S
log
→
Aut(G).
Moreover,
it
follows
imme-
diately
from
the
well-known
structure
of
admissible
coverings
at
nodes
[cf.,
e.g.,
[Mzk1],
§3.23]
that
this
homomorphism
factors
through
I
S
Σ
log
,
hence
determines
a
natural
continuous
homomorphism
ρ
I
:
I
S
Σ
log
→
Aut(G).
Also,
we
recall
that
each
vertex
v
(respectively,
edge
e)
of
G
determines
a(n)
verticial
subgroup
Π
v
⊆
Π
G
(respectively,
edge-like
subgroup
Π
e
⊆
Π
G
),
which
is
well-defined
up
to
conjugation
—
cf.
[Mzk13],
Definition
1.1,
(ii).
Here,
the
edge-like
subgroups
Π
e
may
be
either
nodal
or
cuspidal,
depending
on
whether
e
corresponds
to
a
node
or
to
a
cusp.
If
an
edge
e
corresponds
to
a
node
(respectively,
cusp),
then
we
shall
simply
say
that
e
“is”
a
node
(respectively,
cusp).
(iii)
Let
e
be
a
node
of
X.
Write
M
e
for
the
stalk
of
the
characteristic
sheaf
of
the
log
scheme
X
log
at
e;
M
S
for
the
stalk
of
the
characteristic
sheaf
of
the
log
scheme
S
log
at
the
tautological
S-valued
point
of
S.
Thus,
M
S
∼
=
N;
we
have
a
natural
inclusion
M
S
→
M
e
,
with
respect
to
which
we
shall
often
[by
abuse
of
notation]
identify
M
S
with
its
image
in
M
e
.
Write
σ
∈
M
e
for
the
unique
generator
of
[the
image
of]
M
S
.
Then
there
exist
elements
ξ,
η
∈
M
e
satisfying
the
relation
ξ
+
η
=
i
e
·
σ
for
some
positive
integer
i
e
,
which
we
shall
refer
to
as
the
index
of
the
node
e,
such
that
M
e
is
generated
by
ξ,
η,
σ.
Also,
we
shall
write
i
Σ
e
for
the
largest
positive
integer
j
such
that
i
e
/j
is
a
product
of
primes
∈
Σ
and
refer
to
i
Σ
e
as
the
Σ-index
of
the
node
e.
One
verifies
easily
that
the
set
of
elements
{ξ,
η}
of
M
e
may
be
characterized
intrisically
as
the
set
of
elements
θ
∈
M
e
\M
S
such
that
any
relation
θ
=
n
·
θ
+
θ
for
n
a
positive
integer,
θ
∈
M
e
\M
S
,
θ
∈
M
S
implies
that
n
=
1,
θ
=
0.
In
particular,
i
e
,
i
Σ
e
are
well-defined
and
depend
only
on
the
isomorphism
class
of
the
pair
consisting
of
the
monoid
M
e
and
the
submonoid
⊆
M
e
given
by
the
image
of
M
S
.
Remark
1.1.1.
Of
course,
in
Example
1.1,
it
is
not
necessary
to
assume
that
k
is
separably
closed
[cf.
[Mzk13],
Example
2.5].
If
k
is
not
separably
closed,
then
one
must
also
contend
with
the
action
of
the
absolute
Galois
group
of
k.
More
generally,
for
the
theory
of
the
present
§1,
it
is
not
even
necessary
to
assume
that
an
“additional
profinite
group”
acting
on
G
arises
“from
scheme
theory”.
It
is
this
point
of
view
that
formed
the
motivation
for
Definition
1.2
below.
Definition
1.2.
In
the
notation
of
Example
1.1:
(i)
Let
ρ
H
:
H
→
Aut(G)
(⊆
Out(Π
G
))
be
a
continuous
homomorphism
of
profinite
groups;
suppose
that
X
log
is
nonsingular
[i.e.,
has
no
nodes].
Then
we
shall
refer
to
as
a
[pro-Σ]
PSC-extension
[i.e.,
“pointed
stable
curve
extension”]
any
extension
of
profinite
groups
that
is
isomorphic
—
via
an
isomorphism
which
we
shall
refer
to
as
the
“structure
of
[pro-Σ]
PSC-extension”
—
to
an
extension
of
the
form
out
def
1
→
Π
G
→
Π
H
=
(Π
G
H)
→
H
→
1
10
SHINICHI
MOCHIZUKI
out
[cf.
§0
for
more
on
the
notation
“
”],
which
we
shall
refer
to
as
the
PSC-
extension
associated
to
the
construction
data
(X
log
→
S
log
,
Σ,
G,
ρ
H
).
In
this
sit-
uation,
each
[necessarily
cuspidal]
edge
e
of
G
determines
[up
to
conjugation
in
def
Π
G
]
a
subgroup
Π
e
⊆
Π
G
,
whose
normalizer
D
e
=
N
Π
H
(Π
e
)
in
Π
H
we
shall
refer
to
as
the
decomposition
group
associated
to
the
cusp
e;
we
shall
refer
to
def
I
e
=
Π
e
=
D
e
Π
G
⊆
D
e
[cf.
[Mzk13],
Proposition
1.2,
(ii)]
as
the
inertia
group
associated
to
the
cusp
e.
Finally,
we
shall
apply
the
terminology
applied
to
objects
associated
to
1
→
Π
G
→
Π
H
→
H
→
1
to
the
objects
associated
to
an
arbitrary
PSC-extension
via
its
“structure
of
PSC-extension”
isomorphism.
(ii)
Let
ρ
H
:
H
→
Aut(G)
(⊆
Out(Π
G
))
be
a
continuous
homomorphism
of
profinite
groups;
ι
:
I
S
Σ
log
→
H
a
continuous
injection
of
profinite
groups
with
normal
image
such
that
ρ
H
◦
ι
=
ρ
I
.
Suppose
that
X
log
is
arbitrary
[i.e.,
X
may
be
singular
or
nonsingular].
Then
we
shall
refer
to
as
a
[pro-Σ]
DPSC-extension
[i.e.,
“degenerating
pointed
stable
curve
extension”]
any
extension
of
profinite
groups
that
is
isomorphic
—
via
an
isomorphism
which
we
shall
refer
to
as
the
“structure
of
[pro-Σ]
DPSC-extension”
—
to
an
extension
of
the
form
def
out
1
→
Π
G
→
Π
H
=
(Π
G
H)
→
H
→
1
—
which
we
shall
refer
to
as
the
DPSC-extension
associated
to
the
construction
data
(X
log
→
S
log
,
Σ,
G,
ρ
H
,
ι).
In
this
situation,
we
shall
refer
to
the
image
I
⊆
H
of
ι
as
the
inertia
subgroup
of
H
and
to
the
extension
def
out
1
→
Π
G
→
Π
I
=
(Π
G
I)
→
I
→
1
[so
Π
I
=
Π
H
×
H
I
⊆
Π
H
]
as
the
[pro-Σ]
IPSC-extension
[i.e.,
“inertial
pointed
stable
curve
extension”]
associated
to
the
construction
data
(X
log
→
S
log
,
Σ,
G,
ρ
H
,
ι);
each
vertex
v
(respectively,
edge
e)
of
G
determines
[up
to
conjugation
in
Π
G
]
a
subgroup
Π
v
⊆
Π
G
(respectively,
Π
e
⊆
Π
G
),
whose
normalizer
def
def
D
v
=
N
Π
H
(Π
v
)
(respectively,
D
e
=
N
Π
H
(Π
e
))
in
Π
H
we
shall
refer
to
as
the
decomposition
group
associated
to
v
(respectively,
e);
for
v
arbitrary
(respectively,
e
a
node),
we
shall
refer
to
the
centralizer
def
def
I
v
=
Z
Π
I
(Π
v
)
⊆
D
v
(respectively,
I
e
=
Z
Π
I
(Π
e
)
⊆
D
e
)
as
the
inertia
group
associated
to
v
(respectively,
e).
If
e
is
a
cusp
of
G,
then
we
shall
def
refer
to
I
e
=
Π
e
=
D
e
Π
G
⊆
D
e
[cf.
[Mzk13],
Proposition
1.2,
(ii)]
as
the
inertia
group
associated
to
the
cusp
e.
Finally,
we
shall
apply
the
terminology
applied
to
objects
associated
to
1
→
Π
G
→
Π
H
→
H
→
1
to
the
objects
associated
to
an
arbitrary
DPSC-extension
via
its
“structure
of
DPSC-extension”
isomorphism.
Remark
1.2.1.
Note
that
in
the
situation
of
Definition
1.2,
(i)
(respectively,
(ii)),
any
open
subgroup
of
Π
H
[equipped
with
the
induced
extension
structure]
admits
a
structure
of
[pro-Σ]
PSC-extension
(respectively,
DPSC-extension)
for
appropriate
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
11
construction
data
that
may
be
derived
from
the
original
construction
data.
On
the
other
hand,
in
the
situation
of
Definition
1.2,
(ii),
given
an
open
subgroup
of
Π
I
[equipped
with
the
induced
extension
structure],
in
order
to
endow
this
open
subgroup
with
a
structure
of
IPSC-extension,
it
may
be
necessary
—
even
if,
for
instance,
this
open
subgroup
of
Π
I
surjects
onto
I
—
to
replace
the
inertia
subgroup
I
of
H
by
some
open
subgroup
of
I
[i.e.,
in
effect,
to
replace
the
given
open
subgroup
of
Π
I
with
the
intersection
of
this
given
open
subgroup
with
the
inverse
image
in
Π
I
of
some
open
subgroup
of
I].
Such
replacements
may
be
regarded
as
a
sort
of
abstract
group-theoretic
analogue
of
the
operation
of
passing
to
a
finite
extension
of
a
discretely
valued
field
in
order
to
achieve
a
situation
in
which
a
given
hyperbolic
curve
over
the
original
field
has
stable
reduction.
Remark
1.2.2.
Note
that
in
the
situation
of
Definition
1.2,
(ii),
the
inertia
subgroup
I
⊆
H
is
not
intrinsically
determined
in
the
sense
that
any
open
subgroup
of
I
may
also
serve
as
the
inertia
subgroup
of
H
—
cf.
the
replacement
operation
discussed
in
Remark
1.2.1.
Remark
1.2.3.
Recall
that
for
l
∈
Σ,
one
may
construct
directly
from
G
a
pro-l
cyclotomic
character
χ
l
:
Aut(G)
→
Z
×
l
[cf.
[Mzk13],
Lemma
2.1].
In
particular,
any
ρ
H
as
in
Definition
1.2,
(i),
(ii),
determines
a
pro-l
cyclotomic
character
χ
l
|
H
:
H
→
Z
×
l
.
The
action
ρ
H
is
called
l-cyclotomically
full
[cf.
[Mzk13],
Definition
2.3,
(ii)]
if
the
image
of
χ
l
|
H
is
open.
We
shall
also
apply
this
terminology
“l-
cyclotomically
full”
to
the
corresponding
PSC-,
DPSC-extensions.
In
fact,
it
follows
immediately
from
the
first
portion
of
[Mzk13],
Proposition
2.4,
(iv),
that
the
issue
of
whether
or
not
ρ
H
is
l-cyclotomically
full
depends
only
on
the
outer
representation
H
→
Out(Π
G
)
of
H
on
Π
G
determined
by
ρ
H
.
Proposition
1.3.
(Basic
Properties
of
Inertia
and
Decomposition
Groups)
In
the
notation
of
Definition
1.2,
(ii):
Σ
.
(i)
If
e
is
a
cusp
of
G,
then
as
abstract
profinite
groups,
I
e
∼
=
Z
(ii)
If
e
is
a
node
of
G,
then
we
have
a
natural
exact
sequence
1
→
Π
e
→
Σ
×
Z
Σ
.
If
e
abuts
to
vertices
v,
v
,
I
e
→
I
→
1;
as
abstract
profinite
groups,
I
e
∼
=
Z
then
[for
appropriate
choices
of
conjugates
of
the
various
inertia
groups
involved]
we
have
inclusions
I
v
,
I
v
⊆
I
e
,
and
the
natural
morphism
I
v
×
I
v
→
I
e
is
an
open
injective
homomorphism,
with
image
of
index
equal
to
i
Σ
e
.
∼
(iii)
If
v
is
a
vertex
of
G,
then
we
have
a
natural
isomorphism
I
v
→
I;
Σ
.
If
e
is
a
cusp
that
D
v
Π
I
=
I
v
×
Π
v
;
as
abstract
profinite
groups,
I
v
∼
=
Z
abuts
to
v,
then
[for
appropriate
choices
of
conjugates
of
the
various
inertia
and
decomposition
groups
involved]
we
have
inclusions
I
e
,
I
v
⊆
D
e
Π
I
,
and
the
nat-
ural
morphism
I
e
×
I
v
→
D
e
Π
I
is
an
isomorphism;
in
particular,
as
abstract
∼
Σ
Σ
profinite
groups,
D
e
Π
I
=
Z
×
Z
,
and
we
have
a
natural
exact
sequence
1
→
I
e
→
D
e
Π
I
→
I
→
1.
(iv)
Let
v,
v
be
vertices
of
G.
If
D
v
D
v
Π
I
=
{1},
then
one
of
the
following
three
[mutually
exclusive]
properties
holds:
(1)
v
=
v
;
(2)
v
and
v
are
12
SHINICHI
MOCHIZUKI
distinct,
but
adjacent
[i.e.,
there
exists
a
node
e
that
abuts
to
v,
v
];
(3)
v
and
v
are
distinct
and
non-adjacent,
but
there
exists
a
vertex
v
=
v,
v
of
G
such
that
v
is
adjacent
to
v
and
v
.
Moreover,
in
the
situation
of
(2),
we
have
I
v
I
v
=
{1},
[for
appropriate
choices
of
conjugates
of
the
various
inertia
and
decomposition
D
Π
Π
v
=
=
I
;
in
the
situation
of
(3),
we
have
Π
groups
involved]
D
v
v
I
e
v
of
conjugates
of
I
v
I
v
=
I
v
I
v
=
I
v
I
v
=
{1},
[for
appropriate
choices
the
various
inertia
and
decomposition
groups
involved]
D
v
D
v
Π
I
=
I
v
.
In
particular,
I
v
I
v
=
{1}
implies
that
v
=
v
.
(v)
Let
v
be
a
vertex
of
G.
Then
D
v
=
C
Π
H
(I
v
)
=
N
Π
H
(I
v
)
is
commen-
surably
terminal
in
Π
H
;
D
v
Π
I
=
C
Π
I
(I
v
)
=
N
Π
I
(I
v
)
=
Z
Π
I
(I
v
)
is
com-
mensurably
terminal
in
Π
I
;
D
v
Π
G
=
Π
v
is
commensurably
terminal
in
Π
G
.
(vi)
Let
v
be
a
vertex
of
G.
Then
the
image
of
D
v
in
H
is
open;
on
the
other
hand,
if
G
has
more
than
one
vertex
[i.e.,
the
curve
X
is
singular],
then
D
v
is
not
open
in
Π
H
.
=
N
Π
H
(Π
e
)
is
commensu-
(vii)
Let
e
be
an
edge
of
G.
Then
D
e
=
C
Π
H
(Π
e
)
rably
terminal
in
Π
H
.
If
e
is
a
node,
then
I
e
=
D
e
Π
I
.
(viii)
Let
e,
e
be
edges
of
G.
If
D
e
D
e
Π
I
=
{1},
then
one
of
the
following
two
[mutually
exclusive]
properties
holds:
(1)
e
=
e
;
(2)
e
and
e
are
distinct,
but
abut
to
the
same
vertex
v,
and
D
e
D
e
Π
G
=
{1}.
Moreover,
in
the
situation
of
(2),
[for
appropriate
choices
of
conjugates
of
the
various
inertia
and
decomposition
groups
involved]
we
have
I
v
=
D
e
D
e
Π
I
.
(ix)
Let
e
be
an
edge
of
G.
Then
the
image
of
D
e
in
H
is
open,
but
D
e
is
not
open
in
Π
H
.
(x)
Let
τ
I
:
I
→
Π
I
be
the
[outer]
homomorphism
that
arises
[by
functoriality!]
from
a
“log
point”
τ
S
∈
X
log
(S
log
).
Let
us
call
τ
I
non-verticial
(respectively,
non-edge-like)
if
τ
I
(I)
is
not
contained
in
I
v
(respectively,
I
e
)
for
any
vertex
v
(respectively,
edge
e)
of
G.
Then
if
τ
I
is
non-verticial
and
non-edge-like,
then
the
image
of
τ
S
is
the
unique
cusp
e
τ
of
X
such
that
[for
an
appropriate
choice
of
conjugate
of
D
e
τ
]
τ
I
(I)
⊆
D
e
τ
.
Now
suppose
that
the
image
of
τ
S
is
not
a
cusp.
Then
τ
I
satisfies
the
condition
τ
I
(I)
=
I
v
τ
for
some
vertex
v
τ
of
G
[and
an
appropriate
choice
of
conjugate
of
I
v
τ
]
if
and
only
if
the
image
of
τ
S
is
a
non-nodal
point
of
the
irreducible
component
of
X
corresponding
to
v
τ
;
τ
I
is
non-verticial
and
satisfies
the
condition
τ
I
(I)
⊆
I
e
τ
for
some
node
e
τ
of
G
[and
an
appropriate
choice
of
conjugate
of
I
e
τ
]
if
and
only
if
the
image
of
τ
S
is
the
node
of
X
corresponding
to
e
τ
.
Proof.
Assertion
(i)
follows
immediately
from
the
definitions.
Next,
we
consider
assertion
(ii).
Write
ν
(
∼
=
S)
for
the
closed
subscheme
of
X
determined
by
the
node
of
X
corresponding
to
e;
ν
log
for
the
result
of
equipping
ν
with
the
log
structure
def
pulled
back
from
X
log
.
Thus,
we
obtain
a
natural
[outer]
homomorphism
Π
ν
=
π
1
(ν
log
)
→
π
1
(X
log
)
=
Π
X
log
Π
I
.
Now
[in
the
notation
of
Example
1.1,
(iii)]
one
computes
easily
[by
considering
the
Galois
groups
of
the
various
Kummer
log
étale
∼
coverings
of
ν
log
]
that
we
have
natural
isomorphisms
Π
ν
→
Hom(M
e
gp
⊗
Q/Z,
k
×
),
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
13
∼
I
S
log
→
Hom(M
S
gp
⊗
Q/Z,
k
×
)
[where
“gp”
denotes
the
groupification
of
a
monoid].
Moreover,
if
we
write
Π
ν
Π
Σ
ν
for
the
maximal
pro-Σ
quotient
of
Π
ν
,
then
one
verifies
immediately
that
the
isomorphisms
induced
on
maximal
pro-Σ
quotients
by
these
natural
isomorphisms
are
compatible,
relative
to
the
surjection
∼
gp
gp
×
×
Σ
Σ
∼
Σ
(Π
Σ
ν
→
)
Hom(M
e
⊗
Q/Z,
k
)
⊗
Z
Hom(M
S
⊗
Q/Z,
k
)
⊗
Z
(
→
I
S
log
)
Σ
induced
by
the
inclusion
M
S
→
M
e
,
with
the
morphism
Π
Σ
ν
→
I
S
log
induced
by
∼
gp
Σ
.
the
composite
morphism
Π
ν
→
Π
I
I
→
I
S
Σ
log
∼
=
Hom(M
S
⊗
Q/Z,
k
×
)
⊗
Z
Σ
The
kernel
of
this
surjection
Π
Σ
ν
I
S
log
may
be
identified
with
the
profinite
group
gp
Σ
,
and
one
verifies
immediately
[from
the
definition
Hom(M
e
gp
/M
S
⊗
Q/Z,
k
×
)
⊗
Z
of
G]
that
this
kernel
maps
isomorphically
onto
Π
e
⊆
Π
I
.
In
particular,
it
follows
that
we
obtain
an
injection
Π
Σ
ν
→
Π
I
whose
image
contains
Π
e
and
surjects
onto
I.
Σ
it
follows
that
the
image
Im(Π
Σ
Since
Π
ν
is
abelian,
ν
)
of
this
injection
is
contained
in
I
e
;
since
I
e
Π
G
=
Π
e
[cf.
[Mzk13],
Proposition
1.2,
(ii)],
we
thus
conclude
that
Im(Π
Σ
ν
)
=
I
e
.
Now
it
follows
immediately
from
the
definitions
that
I
v
,
I
v
⊆
I
e
;
moreover,
one
computes
immediately
that
[in
the
notation
of
Example
1.1,
(iii)]
the
Σ
subgroups
I
v
,
I
v
⊆
I
e
correspond
to
the
subgroups
of
Hom(M
e
gp
⊗
Q/Z,
k
×
)
⊗
Z
consisting
of
homomorphisms
that
vanish
on
ξ,
η,
respectively.
Now
the
various
assertions
contained
in
the
statement
of
assertion
(ii)
follow
immediately.
This
completes
the
proof
of
assertion
(ii).
Next,
we
consider
assertion
(iii).
Since
Π
v
is
slim
[cf.,
e.g.,
[Mzk13],
Remark
1.1.3]
and
commensurably
terminal
in
Π
G
[cf.
[Mzk13],
Proposition
1.2,
(ii)],
it
follows
that
D
v
Π
G
=
Π
v
and
I
v
Π
G
=
{1},
so
we
obtain
a
natural
injection
I
v
→
I.
The
fact
that
this
injection
is,
in
fact,
surjective
is
immediate
from
the
definitions
when
X
is
smooth
over
k
and
follows
from
the
computation
of
“I
v
”
performed
in
the
proof
of
assertion
(ii)
when
X
is
singular.
Next,
let
us
observe
[by
definition!]
with
Π
v
,
we
obtain
a
natural
morphism
that
since
I
v
commutes
I
v
×
Π
v
→
D
v
Π
I
,
which
is
both
injective
[since
I
v
Π
v
=
{1}]
and
surjective
∼
[cf.
the
isomorphism
I
v
→
I;
the
fact
that
D
v
Π
G
=
Π
v
].
Now
suppose
that
e
is
a
cusp
that
abuts
to
v.
Then
[for
appropriate
choices
of
conjugates]
it
follows
Π
,
I
⊆
D
,
immediately
from
the
definitions
that
we
have
inclusions
I
e
v
e
I
and
that
I
e
commutes
with
I
v
.
Note,
moreover,
that
D
e
Π
G
=
I
e
[cf.
[Mzk13],
Proposition
1.2,
(ii)].
Thus,
the
fact
that
the
natural
projection
I
v
→
I
is
an
isomorphism
implies
that
we
have
a
natural
exact
sequence
1
→
I
e
→
D
e
Π
I
→
I
→
1,
and
that
the
natural
morphism
I
e
×
I
v
→
D
e
Π
I
is
an
isomorphism.
This
completes
the
proof
of
assertion
(iii).
Next,
we
consider
assertion
(vii).
Since
D
e
Π
G
=
Π
e
[cf.
[Mzk13],
Proposi-
tion
1.2,
(ii)],
it
follows
that
D
e
⊆
C
Π
H
(D
e
)
⊆
C
Π
H
(Π
e
);
on
the
other
hand,
by
[Mzk13],
Proposition
1.2,
(i),
it
follows
that
C
Π
H
(Π
e
)
=
N
Π
H
(Π
e
)
(=
D
e
);
thus,
D
e
=
C
Π
H
(D
e
)
=
C
Π
H
(Π
e
)
=
N
Π
H
(Π
e
),
as
desired.
Now
it
remains
only
to
con-
sider
the
case
where
e
is
a
node.
In
this
case,
since
I
e
is
abelian
[cf.
assertion
(ii)],
it
follows
that
I
e
⊆
Z
Π
I
(I
e
)
⊆
C
Π
I
(I
e
)
⊆
C
Π
I
(I
e
Π
G
)
=
C
Π
I
(Π
e
);
thus,
the
fact
that
D
e
Π
I
=
C
Π
I
(Π
e
)
=
C
Π
I
(I
e
)
=
I
e
follows
from
the
fact
that
I
e
surjects
onto
I
[cf.
assertion
(ii)],
together
with
the
commensurable
terminality
of
Π
e
in
Π
G
[cf.
[Mzk13],
Proposition
1.2,
(ii)].
This
completes
the
proof
of
assertion
(vii).
Next,
we
consider
assertion
(iv).
Suppose
that
(2)
holds.
Then
it
follows
from
assertions
(ii),
(iii)
[and
the
definitions]
that
I
v
I
v
=
{1},
I
e
=
Π
e
·
I
v
=
14
SHINICHI
MOCHIZUKI
Π
e
·
I
v
⊆
D
v
D
v
Π
I
⊆
Z
Π
I
(I
v
×
I
v
)
⊆
C
Π
I
(I
e
).
On
the
other
hand,
by
assertion
(vii),
we
have
C
Π
I
(I
e
)
=
I
e
.
But
this
implies
that
D
v
D
v
Π
I
=
I
e
.
Next,
suppose
that
(3)
holds,
and
that
Π
v
Π
v
=
{1}.
Then
it
follows
immediately
from
our
discussion
of
the
situation
in
which
(2)
holds
[cf.
also
assertion
(iii)]
that
Σ
∼
Π
I
→
I
[so
I
v
=
D
v
D
v
Π
I
],
I
v
I
v
⊆
I
v
I
v
=
(
Z
=)
I
v
⊆
D
v
D
v
I
v
I
v
=
{1}.
Thus,
to
complete
the
proof
of
assertion
(iv),
it
suffices
to
verify
—
under
the
assumption
that
(1)
and
(2)
are
false!
—
that
we
have
an
equality
(D
v
D
v
Π
I
⊇)
Π
v
Π
v
=
{1},
and
that
(3)
holds.
Write
C
v
,
C
v
for
the
irreducible
components
of
X
corresponding
to
v,
v
.
Suppose
that
both
(1)
and
(2)
are
false.
Recall
that
Π
G
[cf.
[MT],
Remark
1.2.2]
Σ
),
hence
also
Π
I
,
are
torsion-free.
Thus,
I
v
×
Π
v
=
D
v
Π
I
[cf.
and
I
(
∼
=
Z
assertion
(iii)]
is
torsion-free,
so
by
replacing
Π
H
by
an
open
subgroup
of
Π
H
[cf.
Remark
1.2.1],
we
may
assume
without
loss
of
generality
that
G
[i.e.,
X
log
]
is
sturdy
[cf.
§0],
and
that
G
is
edge-paired
[cf.
§0].
Also,
by
projecting
to
the
maximal
pro-l
quotients,
for
some
l
∈
Σ,
of
suitable
open
subgroups
[cf.
Remark
1.2.1]
of
the
various
pro-Σ
groups
involved,
one
verifies
immediately
that
we
may
assume
without
loss
of
generality
[for
the
remainder
of
the
proof
of
assertion
(iv)]
that
Σ
=
{l}.
When
l
=
2,
we
may
also
assume
without
loss
of
generality
[by
replacing
Π
H
by
an
open
subgroup
of
Π
H
]
that
G
is
edge-even
[cf.
§0].
Now
I
claim
that
D
v
D
v
Π
G
=
Π
v
Π
v
=
{1}.
Indeed,
suppose
that
Π
v
Π
v
=
{1}.
Then
one
verifies
immediately
that
there
exist
log
admissible
coverings
[cf.
[Mzk1],
§3]
Y
log
→
X
log
×
S
log
T
log
,
corresponding
to
open
subgroups
J
⊆
Π
G
,
which
are
split
over
C
v
[so
Π
v
⊆
J,
Π
v
Π
v
⊆
Π
v
J],
but
determine
arbitrarily
small
neighborhoods
Π
v
J
of
the
identity
element
in
Π
v
.
[Here,
we
note
that
the
existence
of
such
coverings
follows
immediately
from
the
fact
that
X
log
is
edge-paired
for
arbitrary
l
and
edge-even
when
l
=
2.
That
is
to
say,
one
starts
by
constructing
the
covering
over
C
v
in
such
a
way
that
the
ramification
indices
at
the
nodes
and
cusps
of
C
v
are
all
equal;
one
then
extends
the
covering
over
the
irreducible
components
of
X
adjacent
to
v
[by
applying
the
fact
that
X
log
is
edge-paired
for
arbitrary
l
and
edge-even
when
l
=
2
—
cf.
the
discussion
of
§0]
in
such
a
way
that
the
covering
is
unramified
over
the
nodes
of
these
irreducible
components
that
do
not
abut
to
C
v
;
finally,
one
extends
the
covering
to
a
split
covering
over
the
remaining
portion
of
X
[which
includes
C
v
!].]
But
the
existence
of
such
J
implies
that
Π
v
Π
v
=
{1},
a
contradiction.
This
completes
the
proof
of
the
claim.
Thus,
the
natural
projection
D
v
D
v
Π
I
→
I
has
nontrivial
open
image
[since
Σ
=
{l}],
which
we
denote
by
I
v,v
⊆
I.
Moreover,
to
complete
the
proof
of
assertion
(iv),
it
suffices
to
derive
a
contradiction
under
the
assumption
that
(1),
(2),
and
(3)
are
false.
Thus,
for
the
remainder
of
the
proof
of
assertion
(iv),
we
assume
that
(1),
(2),
and
(3)
are
false.
Write
C
v
+
⊆
X
for
the
union
of
C
v
and
the
irreducible
components
of
X
that
are
adjacent
to
C
v
.
We
shall
refer
to
a
vertex
of
G
as
a
C
v
+
-vertex
if
it
corresponds
to
an
irreducible
component
of
C
v
+
.
We
shall
say
that
a
node
e
is
a
bridge
node
if
it
abuts
both
to
a
C
v
+
-vertex
and
to
a
non-C
v
+
-vertex.
Thus,
no
bridge
node
abuts
to
v.
Now
let
us
write
i
v
for
the
least
common
multiple
of
the
indices
i
e
of
the
bridge
nodes
e;
i
Σ
v
for
the
largest
nonnegative
power
of
l
dividing
i
v
.
Let
def
def
Σ
d
=
l
·
i
v
·
[I
:
I
v,v
];
d
Σ
=
l
·
i
Σ
v
·
[I
:
I
v,v
]
[so
d
is
the
largest
positive
power
of
l
dividing
d].
Here,
we
observe
that
for
any
open
subgroup
J
0
⊆
Π
I
such
that
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
D
v
J
0
surjects
onto
I
[cf.
assertion
(iii)],
and
D
v
[I
:
I
v,v
]
=
[(D
v
J
0
)
:
(Π
v
J
0
)
·
(D
v
15
Π
I
⊆
J
0
,
it
holds
that
D
v
Π
I
)]
J
0
].
Thus,
it
suffices
to
construct
[where
we
note
that
D
v
D
v
Π
I
⊆
D
v
onto
open
subgroups
J
⊆
J
0
⊆
Π
I
such
that
D
v
J
0
surjects
I,
Π
v
J
0
⊆
J,
Π
I
Π
I
)
⊆
D
v
J],
but
⊆
J
[which
implies
that
(Π
v
J
0
)
·
(D
v
D
v
and
D
v
[D
v
J
0
:
D
v
J]
>
[I
:
I
v,v
].
To
this
end,
let
us
first
observe
that
the
characteristic
sheaf
of
the
log
scheme
X
log
admits
a
section
ζ
over
X
satisfying
the
following
properties:
(a)
ζ
vanishes
on
the
open
subscheme
of
X
given
by
the
complement
of
C
v
+
[hence,
in
particular,
on
C
v
];
(b)
ζ
coincides
with
i
v
·
σ
∈
M
S
[cf.
the
notation
of
Example
1.1,
(iii)]
at
the
generic
points
of
C
v
+
;
(c)
ζ
coincides
with
either
(i
v
/i
e
)·ξ
∈
M
e
or
(i
v
/i
e
)·η
∈
M
e
[cf.
the
notation
of
Example
1.1,
(iii)]
at
each
bridge
node
e.
[Indeed,
the
existence
of
such
a
section
ζ
follows
immediately
from
the
discussion
of
Example
1.1,
(iii),
together
with
our
definition
of
i
v
.]
Thus,
by
taking
the
inverse
image
of
ζ
in
the
monoid
that
defines
the
log
structure
of
X
log
,
we
obtain
a
line
bundle
L
on
X.
Let
Y
→
X
be
a
finite
étale
cyclic
covering
of
order
a
positive
power
of
l
such
that
L|
Y
has
degree
divisible
by
d
Σ
on
every
irreducible
component
of
Y
,
and
Y
→
X
restricts
to
a
connected
covering
over
every
irreducible
component
def
def
of
X
that
is
=
C
v
[e.g.,
C
v
],
but
splits
over
C
v
;
Y
log
=
X
log
×
X
Y
;
C
w
=
C
v
×
X
Y
;
def
+
=
C
v
+
×
X
Y
.
[Note
that
the
fact
that
such
a
covering
exists
follows
immediately
C
w
from
our
assumption
that
G
is
sturdy.]
Now
let
Z
log
→
Y
log
be
a
log
étale
cyclic
covering
of
degree
d
Σ
satisfying
the
following
properties:
(d)
Z
log
→
Y
log
restricts
to
an
étale
covering
of
schemes
over
the
complement
+
and
splits
over
the
irreducible
components
of
Y
that
lie
over
C
v
of
C
w
[cf.
(a);
the
fact
that
(1),
(2),
and
(3)
are
assumed
to
be
false!];
(e)
Z
log
→
Y
log
is
ramified,
with
ramification
index
d
Σ
/i
Σ
v
,
over
the
generic
+
,
but
induces
the
trivial
extension
of
the
function
field
of
C
w
points
of
C
w
[cf.
(b)];
(f)
for
each
node
f
of
Y
that
lies
over
a
bridge
node
e
of
X,
the
restriction
of
Z
log
→
Y
log
to
the
branch
of
f
that
does
not
abut
to
an
irreducible
+
Σ
is
ramified,
with
ramification
index
d
Σ
·
i
Σ
component
of
C
w
e
/i
v
[cf.
(c)].
Indeed,
to
construct
such
a
covering
Z
log
→
Y
log
,
it
suffices
to
construct
a
covering
+
[which
is
always
possible,
by
the
satisfying
(d),
(f)
over
the
complement
of
C
w
conditions
imposed
on
Y
,
together
with
the
fact
that
(1),
(2),
and
(3)
are
assumed
16
SHINICHI
MOCHIZUKI
to
be
false],
and
then
to
glue
this
covering
to
a
suitable
[i.e.,
such
that
(e)
is
def
+
log
+
)
=
Y
log
×
Y
C
w
[by
an
fs
log
satisfied!]
Kummer
log
étale
covering
of
(C
w
Σ
scheme!]
obtained
by
extracting
a
d
-th
root
of
L|
C
w
+
[cf.
the
divisibility
condition
+
on
the
degrees
of
L
over
the
irreducible
components
of
C
w
].
[Here,
we
regard
the
G
m
-torsor
determined
by
L|
C
w
+
as
a
subsheaf
of
the
monoid
defining
the
log
structure
+
log
)
.]
Now
if
we
write
J
Z
⊆
J
Y
⊆
Π
I
for
the
open
subgroups
defined
by
the
of
(C
w
log
log
log
onto
I;
D
v
Π
I
⊆
J
Y
.
On
coverings
Z
→
Y
→
X
,
then
D
v
J
Y
surjects
the
other
hand,
Π
v
J
Y
⊆
J
Z
[cf.
(e)]
and
D
v
Π
I
⊆
J
Z
[cf.
(d)],
while
[cf.
(e)]
[D
v
J
Y
:
D
v
J
Z
]
=
d
Σ
/i
Σ
v
>
[I
:
I
v,v
]
def
def
[since
d
Σ
=
l
·
i
Σ
v
·
[I
:
I
v,v
]].
Thus,
it
suffices
to
take
J
0
=
J
Y
,
J
=
J
Z
.
This
completes
the
proof
of
assertion
(iv).
Next,
we
consider
assertion
(v).
First,
let
us
observe
that
it
follows
from
assertion
(iv)
[i.e.,
by
applying
assertion
(iv)
to
various
open
subgroups
of
Π
H
,
Π
I
(γ
·
I
v
·
γ
−1
)
=
{1},
then
Π
v
=
—
cf.
also
Remark
1.2.1]
that
if,
for
γ
∈
Π
H
,
I
v
γ
·
Π
v
·
γ
−1
.
Thus,
we
conclude
that
N
Π
H
(I
v
)
⊆
C
Π
H
(I
v
)
⊆
N
Π
H
(Π
v
)
=
D
v
.
On
the
other
hand,
since
[by
definition]
I
v
=
Z
Π
I
(Π
v
),
and
I
is
normal
in
H,
it
follows
(Π
v
)
⊆
N
Π
H
(I
v
),
so
N
Π
H
(I
v
)
=
C
Π
H
(I
v
)
=
D
v
,
as
desired.
In
that
D
v
=
N
Π
H
particular,
D
v
Π
I
=
N
Π
I
(I
v
)
=
C
Π
I
(I
v
).
Next,
let
us
observe
that
D
v
Π
G
=
Π
v
[cf.
[Mzk13],
Proposition
1.2,
(ii)].
Thus,
D
v
⊆
C
Π
H
(D
v
)
⊆
C
Π
H
(Π
v
).
Moreover,
by
[Mzk13],
Proposition
1.2,
(i),
it
follows
that
C
Π
H
(Π
v
)
=
N
Π
H
(Π
v
)
=
D
v
;
thus,
terminal
in
we
conclude
that
D
v
(respectively,
D
v
Π
I
;
D
v
Π
G
)
is
commensurably
assertion
(iii),
we
have
D
v
Π
I
=
I
v
×
Π
v
⊆
Π
H
(respectively,
Π
I
;
Π
G
).
Finally,
by
Z
Π
I
(I
v
)
⊆
N
Π
I
(I
v
)
=
D
v
Π
I
,
so
D
v
Π
I
=
Z
Π
I
(I
v
),
as
desired.
This
completes
the
proof
of
assertion
(v).
Next,
we
consider
assertion
(vi).
The
fact
that
the
image
of
D
v
in
H
is
open
follows
immediately
from
the
fact
that
since
the
semi-graph
G
is
finite,
some
open
subgroup
of
H
necessarily
fixes
v.
On
the
other
hand,
if
G
admits
a
vertex
v
=
v,
then
Π
v
Π
v
is
not
open
in
Π
v
[cf.
[Mzk13],
Proposition
1.2,
(i)];
since
D
v
Π
G
=
Π
v
[cf.
[Mzk13],
Proposition
1.2,
(ii)],
this
implies
that
D
v
is
not
open
in
Π
H
.
This
completes
the
proof
of
assertion
(vi).
Next,
we
consider
assertion
(viii).
First,
we
observe
that
if
property
(2)
holds,
Σ
∼
then
(ii),
(iii),
[for
appropriate
choices
of
conjugates]
(
Z
=)
I
v
⊆
by
assertions
D
e
D
e
Π
I
→
I,
so
I
v
=
D
e
D
e
Π
I
.
Thus,
it
suffices
to
verify
that
ei-
ther
(1)
or
(2)
holds.
Next,
let
us
observe
that,
since,
as
observed
above,
Π
I
,
hence
also
D
e
D
e
Π
I
,
is
torsion-free,
by
projecting
to
the
maximal
pro-l
quo-
tients,
for
some
l
∈
Σ,
of
suitable
open
subgroups
[cf.
Remark
1.2.1]
of
the
var-
ious
pro-Σ
groups
involved,
one
verifies
immediately
we
may
assume
without
loss
of
generality
[for
the
remainder
of
the
proof
of
assertion
(viii)]
that
Σ
=
{l}.
Now
if
D
e
D
e
Π
G
=
{1},
then
[since
Σ
=
{l}]
D
e
D
e
Π
G
is
open
in
Σ
),
D
e
Π
G
(
∼
Σ
)
[cf.
assertions
(ii),
(iii),
(vii)],
so
we
con-
D
e
Π
G
(
∼
=
Z
=
Z
clude
from
[Mzk13],
Proposition
1.2,
(i),
that
e
=
e
.
Thus,
to
complete
the
proof
of
assertion
(viii),
it
suffices
to
derive
a
contradiction
under
the
further
assump-
tion
that
D
e
D
e
Π
G
=
{1},
and
e
and
e
do
not
abut
to
a
common
vertex.
Moreover,
by
replacing
Π
H
by
an
open
subgroup
of
Π
H
[cf.
Remark
1.2.1],
we
may
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
17
assume
without
loss
of
generality
that
G
[i.e.,
X
log
]
is
sturdy
[cf.
§0],
and
that
G
is
edge-paired
[cf.
§0]
for
arbitrary
l
and
edge-even
[cf.
§0]
when
l
=
2.
Now
if,
say,
e
is
a
cusp
that
abuts
to
a
vertex
v,
then
one
verifies
immediately
that
there
exist
log
étale
cyclic
coverings
Y
log
→
X
log
of
degree
an
arbitrarily
large
power
of
l
which
are
totally
ramified
over
e,
but
unramified
over
the
nodes
of
X,
as
well
as
over
the
cusps
of
X
that
abut
to
vertices
=
v.
[Indeed,
the
existence
of
such
coverings
follows
immediately
from
the
fact
that
X
log
is
edge-paired
for
arbitrary
l
and
edge-even
when
l
=
2
—
cf.
the
discussion
of
§0.]
In
particular,
such
coverings
are
unramified
over
e
,
as
well
as
over
the
generic
point
of
the
irreducible
component
of
X
corresponding
to
correspond
v,
hence
to
open
subgroups
J
⊆
Π
I
such
that
D
e
Π
I
⊆
J
D
D
e
Π
I
⊆
J
[so
D
e
e
Π
I
],
and,
moreover,
J
may
be
chosen
so
that
the
subgroup
J
D
e
Π
I
⊆
D
e
Π
I
=
I
e
×
I
v
[cf.
assertion
(iii)]
forms
an
arbitrarily
small
neighborhood
of
I
v
.
Thus,
we
conclude
that
D
e
D
e
Π
I
⊆
I
v
.
vertex
v
,
then
[by
symmetry]
On
the
other
hand,
if
e
is
also
a
cusp
that
abuts
to
a
we
conclude
that
D
e
D
e
Π
I
⊆
I
v
,
hence
that
I
v
I
v
=
{1}.
But,
by
assertion
(iv),
this
implies
that
v
=
v
,
a
contradiction.
Thus,
we
may
assume
that,
say,
e
is
a
node,
so
I
e
=
D
e
Π
I
[cf.
assertion
(vii)].
Write
v
1
,
v
2
for
the
two
distinct
vertices
to
which
e
abuts;
C
1
,
C
2
for
the
def
irreducible
components
of
X
corresponding
to
v
1
,
v
2
;
C
=
C
1
C
2
⊆
X;
U
C
⊆
C
for
the
open
subscheme
obtained
by
removing
the
nodes
that
abut
to
vertices
=
v
1
,
v
2
.
Let
us
refer
to
the
nodes
and
cusps
of
U
C
as
inner,
to
the
nodes
of
X
that
were
removed
from
C
to
obtain
U
C
as
bridge
nodes,
and
to
the
nodes
and
cusps
of
X
which
are
neither
inner
nodes/cusps
nor
bridge
nodes
as
external.
[Thus,
e
is
inner;
e
is
external.]
Observe
that
the
natural
projection
to
I
yields
an
inclusion
D
e
D
e
Π
I
→
I
with
open
image
[since
Σ
=
{l}];
denote
the
image
of
this
inclusion
by
I
C
.
Write
i
C
for
the
least
common
multiple
of
the
indices
i
f
of
the
bridge
nodes
f
;
i
Σ
C
for
the
largest
nonnegative
power
of
l
dividing
i
C
.
Let
def
def
Σ
d
=
l
·
i
C
·
[I
:
I
C
];
d
Σ
=
l
·
i
Σ
C
·
[I
:
I
C
]
[so
d
is
the
largest
positive
power
of
l
dividing
d].
Here,
we
observe
that
[I
:
I
C
]
=
[I
e
:
Π
e
·
(D
e
D
e
Π
I
)]
[cf.
assertion
(ii)].
subgroup
Then
it
suffices
to
construct
an
open
J
⊆
Π
I
such
that
Π
e
⊆
J
and
D
e
Π
I
⊆
J
[which
implies
that
Π
e
·
(D
e
D
e
Π
I
)
⊆
I
e
J],
but
[I
e
:
I
e
J]
>
[I
:
I
C
].
X
log
To
this
end,
let
us
first
observe
that
the
characteristic
sheaf
of
the
log
scheme
admits
a
section
ζ
over
X
satisfying
the
following
properties:
(a)
ζ
vanishes
on
the
open
subscheme
of
X
given
by
the
complement
of
C;
(b)
ζ
coincides
with
i
C
·
σ
∈
M
S
[cf.
the
notation
of
Example
1.1,
(iii)]
at
def
the
generic
points
of
C
=
C
1
C
2
;
(c)
ζ
coincides
with
either
(i
C
/i
f
)
·
ξ
∈
M
f
or
(i
C
/i
f
)
·
η
∈
M
f
[cf.
the
notation
of
Example
1.1,
(iii),
where
we
take
“e”
to
be
f
]
at
each
bridge
node
f
.
18
SHINICHI
MOCHIZUKI
[Indeed,
the
existence
of
such
a
section
ζ
follows
immediately
from
the
discussion
of
Example
1.1,
(iii),
together
with
our
definition
of
i
C
.]
Thus,
by
taking
the
inverse
image
of
ζ
in
the
monoid
that
defines
the
log
structure
of
X
log
,
we
obtain
a
line
bundle
L
on
X.
Let
Y
→
X
be
a
finite
étale
Galois
covering
of
order
a
positive
power
of
l
such
that
L|
Y
has
degree
divisible
by
d
Σ
on
every
irreducible
component
of
Y
,
and
Y
→
X
restricts
to
a
connected
covering
over
every
irreducible
def
component
of
X;
Y
log
=
X
log
×
X
Y
.
[Note
that
the
fact
that
such
a
covering
exists
follows
immediately
from
our
assumption
that
G
is
sturdy.]
Write
C
1
Y
,
C
2
Y
for
the
irreducible
components
of
Y
lying
over
C
1
,
C
2
,
respectively;
we
shall
apply
the
terms
“internal”,
“external”,
and
“bridge”
to
nodes/cusps
of
Y
that
lie
over
such
nodes/cusps
of
X.
Now
let
Z
log
→
Y
log
be
a
log
étale
cyclic
covering
of
degree
d
Σ
satisfying
the
following
properties:
(d)
Z
log
→
Y
log
restricts
to
an
étale
covering
of
schemes
over
the
comple-
def
ment
of
C
Y
=
C
1
Y
C
2
Y
[cf.
(a)],
hence,
in
particular,
over
the
external
nodes/cusps
of
Y
;
(e)
Z
log
→
Y
log
is
ramified,
with
ramification
index
d
Σ
/i
Σ
C
,
over
the
generic
Y
Y
points
of
C
1
,
C
2
,
and,
at
each
internal
node
of
Y
lying
over
e,
determines
a
covering
corresponding
to
an
open
subgroup
of
I
e
that
contains
Π
e
[cf.
(b)];
(f)
for
each
bridge
node
f
of
Y
,
the
restriction
of
Z
log
→
Y
log
to
the
branch
Σ
of
f
that
does
not
abut
to
C
Y
is
ramified,
with
ramification
index
d
Σ
·i
Σ
f
/i
C
[cf.
(c)].
Indeed,
to
construct
such
a
covering
Z
log
→
Y
log
,
it
suffices
to
construct
a
covering
satisfying
(d),
(f)
over
the
complement
of
C
Y
[which
is
always
possible,
by
the
conditions
imposed
on
Y
],
and
then
to
glue
this
covering
to
a
suitable
[i.e.,
such
def
that
(e)
is
satisfied!]
Kummer
log
étale
covering
of
(C
Y
)
log
=
Y
log
×
Y
C
Y
[by
an
fs
log
scheme!]
obtained
by
extracting
a
d
Σ
-th
root
of
L|
C
Y
[cf.
the
divisibility
condition
on
the
degrees
of
L|
C
1
Y
,
L|
C
2
Y
].
[Here,
we
regard
the
G
m
-torsor
determined
by
L|
C
Y
as
a
subsheaf
of
the
monoid
defining
the
log
structure
of
(C
Y
)
log
.]
Now
if
the
open
subgroups
defined
by
the
coverings
Z
log
→
we
write
J
Z
⊆
J
Y
⊆
Π
I
for
Y
log
→
X
log
,
then
I
e
,
D
e
Π
I
⊆
J
Y
.
On
the
other
hand,
Π
e
⊆
J
Z
[cf.
(e)]
and
D
e
Π
I
⊆
J
Z
[cf.
(d)],
while
[cf.
(e)]
[I
e
:
I
e
J
Z
]
=
d
Σ
/i
Σ
C
>
[I
:
I
C
]
def
[since
d
Σ
=
l
·
i
Σ
C
·
[I
:
I
C
]].
Thus,
it
suffices
to
take
J
=
J
Z
.
This
completes
the
proof
of
assertion
(viii).
Next,
we
consider
assertion
(ix).
The
fact
that
the
image
of
D
e
in
H
is
open
fol-
lows
immediately
from
the
fact
that
since
the
semi-graph
G
is
finite,
some
open
sub-
group
of
H
necessarily
fixes
e.
On
the
other
hand,
since
D
e
Π
G
=
N
Π
G
(Π
e
)
=
Π
e
[cf.
[Mzk13],
Proposition
1.2,
(ii)]
is
abelian,
hence
not
open
in
the
slim,
nontrivial
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
19
profinite
group
Π
G
,
it
follows
that
D
e
is
not
open
in
Π
H
.
This
completes
the
proof
of
assertion
(ix).
Finally,
we
consider
assertion
(x).
First,
let
us
observe
that
an
easy
com-
putation
reveals
that
if
the
image
of
τ
S
is
a
non-nodal,
non-cuspidal
point
of
the
irreducible
component
of
X
corresponding
to
a
vertex
v
τ
of
G,
then
τ
I
(I)
=
I
v
τ
.
Next,
let
us
suppose
that
the
image
of
τ
S
is
the
node
of
X
corresponding
to
some
node
e
τ
of
G.
Then
an
easy
computation
[cf.
the
computations
performed
in
the
proof
of
assertion
(ii)]
reveals
that
τ
I
(I)
⊆
I
e
τ
,
but
that
τ
I
(I)
is
not
contained
in
I
v
for
any
vertex
v
to
which
e
τ
abuts.
If,
moreover,
τ
I
(I)
⊆
I
v
for
some
vertex
v
to
which
e
τ
does
not
abut,
then
[since
the
very
existence
of
the
node
e
τ
implies
that
X
is
singular]
there
exists
a
node
e
=
e
τ
that
abuts
to
v,
so
τ
I
(I)
⊆
I
v
⊆
I
e
[cf.
as-
sertion
(ii)];
but
this
implies
that
τ
I
(I)
⊆
I
e
I
e
τ
,
so,
by
assertion
(viii),
it
follows
that
τ
I
(I)
⊆
I
v
for
some
vertex
to
which
both
e
and
e
τ
abut
—
a
contradiction.
Thus,
in
summary,
we
conclude
that
in
this
case,
τ
I
is
non-verticial.
Now
suppose
that
τ
I
is
non-verticial
and
non-edge-like.
Then
the
observations
of
the
preceding
paragraph
imply
that
the
image
of
τ
S
is
a
cusp
of
X.
Write
e
τ
for
the
corresponding
cusp
of
G.
Thus,
one
verifies
immediately
that
τ
I
(I)
⊆
D
e
τ
.
The
uniqueness
of
e
τ
then
follows
from
assertion
(viii)
[and
the
fact
that
τ
I
is
non-
verticial].
Thus,
for
the
remainder
of
the
proof
of
assertion
(x),
we
may
assume
that
the
image
of
τ
S
is
not
a
cusp.
Now
the
remainder
of
assertion
(x)
follows
formally,
in
light
of
what
of
we
have
done
so
far,
from
assertions
(iv),
(viii).
This
completes
the
proof
of
assertion
(x).
Corollary
1.4.
(Graphicity
of
Isomorphisms
of
(D)PSC-Extensions)
Let
l
be
a
prime
number.
For
i
=
1,
2,
let
1
→
Π
G
i
→
Π
H
i
→
H
i
→
1
be
an
l-cyclotomically
full
[cf.
Remark
1.2.3]
DPSC-extension
(respectively,
PSC-
extension),
associated
to
construction
data
(X
i
log
→
S
i
log
,
Σ
i
,
G
i
,
ρ
H
i
,
ι
i
)
(re-
spectively,
(X
i
log
→
S
i
log
,
Σ
i
,
G
i
,
ρ
H
i
))
such
that
l
∈
Σ
i
;
in
the
non-resp’d
case,
write
I
i
⊆
H
i
for
the
inertia
subgroup.
Let
∼
φ
H
:
H
1
→
H
2
;
∼
φ
Π
:
Π
G
1
→
Π
G
2
be
compatible
[i.e.,
with
the
respective
outer
actions
of
H
i
on
Π
G
i
]
isomorphisms
of
profinite
groups;
in
the
non-resp’d
case,
suppose
further
that
φ
H
(I
1
)
=
I
2
.
Then
Σ
1
=
Σ
2
;
φ
Π
is
graphic
[cf.
[Mzk13],
Definition
1.4,
(i)],
i.e.,
arises
from
∼
an
isomorphism
of
semi-graphs
of
anabelioids
G
1
→
G
2
.
Proof.
This
follows
immediately
from
[Mzk13],
Corollary
2.7,
(i),
(iii).
Here,
as
in
the
proof
of
[Mzk13],
Corollary
2.8,
we
first
apply
[Mzk13],
Corollary
2.7,
(i)
[which
suffices
to
complete
the
proof
of
Corollary
1.4
in
the
resp’d
case
and
allows
one
to
reduce
to
the
noncuspidal
case
in
the
non-resp’d
case],
then
apply
[Mzk13],
Corollary
2.7,
(iii),
to
the
compactifications
of
corresponding
sturdy
finite
étale
coverings
of
the
G
i
.
We
are
now
ready
to
define
a
purely
group-theoretic,
combinatorial
analogue
of
the
notion
of
a
stable
polycurve
given
in
[Mzk2],
Definition
4.5.
20
SHINICHI
MOCHIZUKI
Definition
1.5.
We
shall
refer
to
an
extension
of
profinite
groups
as
a
PPSC-
extension
[i.e.,
“poly-PSC-extension”]
if,
for
some
positive
integer
n
and
some
nonempty
set
of
primes
Σ,
it
admits
a
“structure
of
pro-Σ
PPSC-extension
of
dimension
n”.
Here,
for
n
a
positive
integer,
Σ
a
nonempty
set
of
primes,
and
1
→
Δ
→
Π
→
H
→
1
an
extension
of
profinite
groups,
we
define
the
notion
of
a
structure
of
pro-Σ
PPSC-
extension
of
dimension
n
as
follows
[inductively
on
n]:
(i)
A
structure
of
pro-Σ
PPSC-extension
of
dimension
1
on
the
extension
1
→
Δ
→
Π
→
H
→
1
is
defined
to
be
a
structure
of
pro-Σ
PSC-extension.
Suppose
that
the
extension
1
→
Δ
→
Π
→
H
→
1
is
equipped
with
a
structure
of
pro-Σ
PPSC-
extension
of
dimension
1.
Thus,
we
have
an
associated
semi-graph
of
anabelioids
G,
together
with
a
continuous
action
of
H
on
G,
and
a
compatible
isomorphism
∼
Δ
→
Π
G
.
We
define
the
[horizontal]
divisors
of
this
PPSC-extension
to
be
the
cusps
of
the
PSC-extension
1
→
Δ
→
Π
→
H
→
1.
Thus,
each
divisor
c
of
the
PPSC-extension
1
→
Δ
→
Π
→
H
→
1
has
associated
inertia
and
decomposition
groups
I
c
⊆
D
c
⊆
Π
[cf.
Definition
1.2,
(i)].
Moreover,
by
[Mzk13],
Proposition
1.2,
(i),
a
divisor
is
completely
determined
by
[the
conjugacy
class
of]
its
inertia
group,
as
well
as
by
[the
conjugacy
class
of]
its
decomposition
group.
Finally,
we
shall
refer
to
the
extension
1
→
Δ
→
Π
→
H
→
1
[itself]
as
the
fiber
extension
associated
to
the
PPSC-extension
1
→
Δ
→
Π
→
H
→
1
of
dimension
1.
(ii)
A
structure
of
pro-Σ
PPSC-extension
of
dimension
n
+
1
on
the
extension
1
→
Δ
→
Π
→
H
→
1
is
defined
to
be
a
collection
of
data
as
follows:
def
(a)
a
quotient
Π
Π
∗
such
that
Δ
†
=
Ker(Π
Π
∗
)
⊆
Δ;
thus,
the
image
Δ
∗
⊆
Π
∗
of
Δ
in
Π
∗
determines
an
extension
1
→
Δ
∗
→
Π
∗
→
H
→
1
—
which
we
shall
refer
to
as
the
associated
base
extension;
the
subgroup
Δ
†
⊆
Π
determines
an
extension
1
→
Δ
†
→
Π
→
Π
∗
→
1
—
which
we
shall
refer
to
as
the
associated
fiber
extension;
(b)
a
structure
of
pro-Σ
PPSC-extension
of
dimension
n
on
the
base
extension
1
→
Δ
∗
→
Π
∗
→
H
→
1;
(c)
a
structure
of
pro-Σ
PPSC-extension
of
dimension
1
on
the
fiber
extension
1
→
Δ
†
→
Π
→
Π
∗
→
1;
(d)
for
each
base
divisor
[i.e.,
divisor
of
the
base
extension]
c
∗
,
a
structure
of
DPSC-extension
on
the
extension
1
→
Δ
†
→
Π
c
∗
=
Π
×
Π
∗
D
c
∗
→
D
c
∗
→
1
def
—
which
we
shall
refer
to
as
the
extension
at
c
∗
—
which
is
compatible
with
the
PSC-extension
structure
on
the
fiber
extension
[cf.
(c)],
in
the
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
21
sense
that
both
structures
yield
the
same
cuspidal
inertia
subgroups
⊆
Δ
†
;
also,
we
require
that
the
inertia
subgroup
of
D
c
∗
[i.e.,
that
arises
from
this
structure
of
DPSC-extension]
be
equal
to
I
c
∗
.
In
this
situation,
we
shall
refer
to
Σ
as
the
fiber
prime
set
of
the
PPSC-extension
1
→
Δ
→
Π
→
H
→
1;
we
shall
refer
to
as
a
divisor
of
the
PPSC-extension
1
→
Δ
→
Π
→
H
→
1
any
element
of
the
union
of
the
set
of
cusps
—
which
we
shall
refer
to
as
horizontal
divisors
—
of
the
PSC-extension
1
→
Δ
†
→
Π
→
Π
∗
→
1
and,
for
each
base
divisor
c
∗
,
the
set
of
vertices
of
the
DPSC-extension
1
→
Δ
†
→
Π
c
∗
→
D
c
∗
→
1
—
which
we
shall
refer
to
as
vertical
divisors
[lying
over
c
∗
].
Thus,
each
divisor
c
of
the
PPSC-extension
1
→
Δ
→
Π
→
H
→
1
has
associated
inertia
and
decomposition
groups
I
c
⊆
D
c
⊆
Π.
In
particular,
whenever
c
is
vertical
and
lies
over
a
base
divisor
c
∗
,
we
have
I
c
⊆
D
c
⊆
Π
c
∗
.
Remark
1.5.1.
Thus,
[the
collection
of
fiber
extensions
arising
from]
any
struc-
ture
of
PPSC-extension
of
dimension
n
on
an
extension
1
→
Δ
→
Π
→
H
→
1
determine
two
compatible
sequences
of
surjections
def
def
def
def
Δ
n
=
Δ
Δ
n−1
.
.
.
Δ
1
Δ
0
=
{1}
Π
n
=
Π
Π
n−1
.
.
.
Π
1
Π
0
=
H
such
that
each
[extension
determined
by
a]
surjection
Π
m
Π
m−1
,
for
m
=
1,
.
.
.
,
n,
is
a
fiber
extension
[hence
equipped
with
a
structure
of
PSC-extension];
Δ
m
=
Ker(Π
m
H).
If
c
=
c
n
is
a
divisor
of
[the
extension
determined
by]
Π
=
Π
n
,
then
[cf.
Definition
1.5,
(ii)]
there
exists
a
uniquely
determined
sequence
of
divisors
c
n
→
c
n−1
→
.
.
.
→
c
n
c
−1
→
c
n
c
—
where
n
c
≤
n
is
a
positive
integer;
for
m
=
n
c
,
.
.
.
,
n,
c
m
is
a
divisor
of
Π
m
;
c
n
c
is
a
horizontal
divisor;
the
notation
“
→”
denotes
the
relation
of
“lying
over”
[so
c
m+1
is
a
vertical
divisor
that
lies
over
c
m
,
for
n
c
≤
m
<
n]
—
together
with
sequences
of
[conjugacies
classes
of
]
inertia
and
decomposition
groups
I
c
n
→
I
c
n−1
→
.
.
.
→
I
c
nc
−1
→
I
c
nc
D
c
n
→
D
c
n−1
→
.
.
.
→
D
c
nc
−1
→
D
c
nc
[i.e.,
for
n
c
≤
m
<
n,
I
c
m+1
⊆
Π
m+1
maps
into
I
c
m
⊆
Π
m
,
and
D
c
m+1
⊆
Π
m+1
maps
into
D
c
m
⊆
Π
m
].
Remark
1.5.2.
Let
1
→
Δ
→
Π
→
H
→
1
be
a
PPSC-extension
of
dimension
n
[where
n
is
a
positive
integer].
Then
one
verifies
immediately
that
if
Π
•
⊆
Π
is
any
open
subgroup
of
Π,
then
there
exists
an
open
subgroup
Π
••
of
Π
•
that
[when
equipped
with
the
induced
extension
structure]
admits
a
structure
of
PPSC-
extension
of
dimension
n
—
cf.
Remark
1.2.1.
Here,
we
note
that
one
must,
in
general,
pass
to
“some
open
subgroup”
Π
••
of
Π
•
in
order
to
achieve
a
situation
in
which
all
of
the
fiber
[PSC-]extensions
have
“stable
reduction”
[cf.
Remark
1.2.1;
Definition
1.5,
(ii),
(d)].
22
SHINICHI
MOCHIZUKI
Remark
1.5.3.
For
l
a
prime
number,
we
shall
say
that
a
PPSC-extension
1
→
Δ
→
Π
→
H
→
1
of
dimension
n
is
l-cyclotomically
full
if
each
of
its
n
associated
fiber
extensions
[cf.
Remark
1.5.1]
is
l-cyclotomically
full
as
a
PSC-extension
[cf.
Remark
1.2.3].
Thus,
it
follows
immediately
from
the
final
portion
of
Remark
1.2.3
that
the
issue
of
whether
or
not
the
PPSC-extension
1
→
Δ
→
Π
→
H
→
1
is
l-cyclotomically
full
depends
only
on
the
sequence
of
surjections
of
profinite
groups
Π
n
Π
n−1
.
.
.
Π
1
Π
0
[cf.
Remark
1.5.1].
Remark
1.5.4.
def
Let
k
be
a
field;
k
a
separable
closure
of
k;
G
k
=
Gal(k/k);
def
S
=
Spec(k);
Z
log
→
S
the
log
scheme
determined
by
a
stable
polycurve
over
S
—
i.e.,
Z
log
admits
a
successive
fibration
by
generically
smooth
stable
log
curves
[cf.
[Mzk2],
Definition
def
4.5,
for
more
details];
U
Z
⊆
Z
the
interior
of
Z
log
;
D
Z
=
Z\U
Z
[with
the
reduced
induced
structure];
n
the
dimension
of
the
scheme
Z;
def
def
1
→
Δ
Z
=
π
1
(U
Z
×
k
k)
→
Π
Z
=
π
1
(U
Z
)
→
G
k
→
1
the
exact
sequence
of
étale
fundamental
groups
[well-defined
up
to
inner
automor-
phism]
associated
to
the
structure
morphism
U
Z
→
S.
Then
by
repeated
appli-
cation
of
the
discussion
of
Example
1.1
to
the
fibers
of
the
successive
fibration
[mentioned
above]
of
Z
log
by
stable
log
curves,
one
verifies
immediately
that:
(i)
If
k
is
of
characteristic
zero,
then
the
structure
of
stable
polycurve
on
Z
log
determines
a
structure
of
profinite
PPSC-extension
of
dimension
n
on
the
extension
1
→
Δ
Z
→
Π
Z
→
G
k
→
1.
Moreover,
one
verifies
immediately
that:
(ii)
In
the
situation
of
(i),
the
divisors
of
Π
Z
[in
the
sense
of
Definition
1.5]
are
in
natural
bijective
correspondence
with
the
irreducible
divisors
of
D
Z
in
a
fashion
that
is
compatible
with
the
inertia
and
decomposition
groups
of
divisors
of
Π
Z
[in
the
sense
of
Definition
1.5]
and
of
irreducible
divisors
of
D
Z
[in
the
usual
sense].
Finally,
whether
or
not
k
is
of
characteristic
zero,
depending
on
the
structure
of
Z
log
[cf.,
e.g.,
Corollary
1.10
below],
various
quotients
of
the
extension
1
→
Δ
Z
→
Π
Z
→
G
k
→
1
may
be
equipped
with
a
structure
of
pro-Σ
PPSC-extension
[induced
by
the
structure
of
stable
polycurve
on
Z],
for
various
nonempty
sets
of
prime
numbers
Σ
that
are
not
equal
to
the
set
of
all
prime
numbers;
a
similar
observation
to
(ii)
concerning
a
natural
bijective
correspondence
of
“divisors”
then
applies
to
such
quotients.
When
considering
such
quotients
1
→
Δ
→
Π
→
H
→
1
of
the
extension
1
→
Δ
Z
→
Π
Z
→
G
k
→
1,
it
is
useful
to
observe
that
the
slimness
of
Δ
[cf.
Proposition
1.6,
(i),
below]
implies
that
such
a
quotient
Π
Z
Π
is
completely
determined
by
the
induced
quotients
Δ
Z
Δ,
G
k
H
[cf.
the
discussion
of
the
out
notation
“
”
in
§0];
we
shall
refer
to
such
a
quotient
1
→
Δ
→
Π
→
H
→
1
as
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
23
a
PPSC-extension
arising
from
Z
log
,
k
—
where
we
write
k
⊆
k
for
the
subfield
fixed
by
Ker(G
k
H).
Proposition
1.6.
(Basic
Properties
of
PPSC-Extensions)
Let
1
→
Δ
→
Π
→
H
→
1
be
a
pro-Σ
PPSC-extension
of
dimension
n
[where
n
is
a
positive
integer];
1
→
Δ
†
→
Π
→
Π
∗
→
1
the
associated
fiber
extension;
c,
c
divisors
of
Π.
Then:
(i)
Δ
is
slim.
In
particular,
if
H
is
slim,
then
so
is
Π.
(ii)
D
c
is
commensurably
terminal
in
Π.
Σ
.
(iii)
We
have:
C
Π
(I
c
)
=
D
c
.
As
abstract
profinite
groups,
I
c
∼
=
Z
(iv)
D
c
is
not
open
in
Π.
The
divisor
c
is
horizontal
if
and
only
if
D
c
projects
to
an
open
subgroup
of
Π
∗
.
If
c
is
vertical
and
lies
over
a
base
divisor
c
∗
,
then
D
c
projects
onto
an
open
subgroup
of
D
c
∗
.
(v)
If
D
c
D
c
is
open
in
D
c
,
D
c
,
then
c
=
c
.
In
particular,
a
divisor
of
Π
is
completely
determined
by
its
associated
decomposition
group.
(vi)
If
I
c
I
c
is
open
in
I
c
,
I
c
,
then
c
=
c
.
In
particular,
a
divisor
of
Π
is
completely
determined
by
its
associated
inertia
group.
Proof.
Assertion
(i)
follows
immediately
from
the
“slimness
of
Π
G
”
discussed
in
Example
1.1,
(ii)
[cf.
Definition
1.5,
(i);
Definition
1.5,
(ii),
(c)].
Next,
we
consider
assertion
(ii).
We
apply
induction
on
n.
If
c
is
horizontal,
then
assertion
(ii)
follows
from
[the
argument
applied
in]
Proposition
1.3,
(vii)
[cf.
also
Definition
1.5,
(ii),
(c)].
If
c
is
vertical,
then
c
lies
over
some
base
divisor
c
∗
,
and
we
are
in
the
situation
of
Definition
1.5,
(ii),
(d).
By
Proposition
1.3,
(vi),
it
follows
that
D
c
surjects
onto
some
open
subgroup
of
D
c
∗
,
hence
that
C
Π
(D
c
)
maps
into
C
Π
∗
(D
c
∗
);
by
the
induction
hypothesis,
C
Π
∗
(D
c
∗
)
=
D
c
∗
,
so
C
Π
(D
c
)
⊆
Π
c
∗
.
Thus,
the
fact
that
C
Π
(D
c
)
=
D
c
follows
from
Proposition
1.3,
(v).
This
completes
the
proof
of
assertion
(ii).
Next,
we
consider
assertion
(iii).
Again
we
apply
induction
on
n.
If
c
is
horizontal,
then
assertion
(iii)
follows
from
[the
argument
applied
in]
Proposition
1.3,
(i),
(vii).
If
c
is
vertical,
then
c
lies
over
some
base
divisor
c
∗
,
and
we
are
in
the
situation
of
Definition
1.5,
(ii),
(d).
By
Proposition
1.3,
(iii)
[cf.
Definition
1.5,
∼
Σ
.
In
particular,
C
Π
(I
c
)
maps
into
(ii),
(d)],
we
have
isomorphisms
I
c
→
I
c
∗
∼
=
Z
C
Π
∗
(I
c
∗
);
by
the
induction
hypothesis,
C
Π
∗
(I
c
∗
)
=
D
c
∗
.
Thus,
C
Π
(I
c
)
⊆
Π
c
∗
,
so
the
fact
that
C
Π
(I
c
)
=
D
c
follows
from
Proposition
1.3,
(v).
This
completes
the
proof
of
assertion
(iii).
Next,
we
consider
assertion
(iv).
Again
we
apply
induction
on
n.
If
c
is
horizontal,
then
by
[the
argument
applied
in]
Proposition
1.3,
(ix),
D
c
is
not
open
in
Π,
but
D
c
projects
to
an
open
subgroup
of
Π
∗
.
If
c
is
vertical,
then
c
lies
over
some
base
divisor
c
∗
,
and
we
are
in
the
situation
of
Definition
1.5,
(ii),
(d);
D
c
⊆
Π
c
∗
.
By
Proposition
1.3,
(vi),
D
c
projects
onto
an
open
subgroup
of
D
c
∗
.
By
the
induction
24
SHINICHI
MOCHIZUKI
hypothesis,
D
c
∗
is
not
open
in
Π
∗
,
so
Π
c
∗
is
not
open
in
Π;
thus,
D
c
is
not
open
in
Π,
and
its
image
in
Π
∗
is
not
open
in
Π
∗
.
This
completes
the
proof
of
assertion
(iv).
Next,
we
consider
assertion
(v).
Again
we
apply
induction
on
n.
By
assertion
(iv),
c
is
horizontal
if
and
only
if
c
is.
If
c,
c
are
horizontal,
then
the
fact
that
c
=
c
follows
from
[Mzk13],
Proposition
1.2,
(i),
(ii).
Thus,
we
may
suppose
that
∗
∗
c,
c
are
vertical
and
lie
over
respective
base
divisors
c
,
(c
)
.
By
assertion
(iv),
it
follows
that
D
c
∗
D
(c
)
∗
is
open
in
D
c
∗
,
D
(c
)
∗
;
by
the
induction
hypothesis,
this
implies
that
c
∗
=
(c
)
∗
.
Thus,
by
intersecting
with
“Π
G
”
[cf.
Proposition
1.3,
(v)]
and
applying
[Mzk13],
Proposition
1.2,
(i),
we
conclude
that
c
=
c
.
This
completes
the
proof
of
assertion
(v).
Finally,
we
observe
that
assertion
(vi)
is
an
immediate
consequence
of
assertions
(iii),
(v).
We
are
now
ready
to
state
and
prove
the
main
result
of
the
present
§1.
Theorem
1.7.
(Graphicity
of
Isomorphisms
of
PPSC-Extensions)
Let
l
be
a
prime
number;
n
a
positive
integer.
For
=
α,
β,
let
Σ
be
a
nonempty
set
of
primes;
1
→
Δ
→
Π
→
H
→
1
an
l-cyclotomically
full
[cf.
Remark
1.5.3]
pro-Σ
PPSC-extension
of
dimension
n;
Π
n
=
Π
Π
n−1
.
.
.
Π
1
Π
0
=
H
def
def
the
sequence
of
successive
fiber
extensions
associated
to
Π
[cf.
Remark
1.5.1].
Let
∼
φ
:
Π
α
→
Π
β
∼
β
be
an
isomorphism
of
profinite
groups
that
induces
isomorphisms
φ
m
:
Π
α
m
→
Π
m
,
for
m
=
0,
1,
.
.
.
,
n
[so
φ
=
φ
n
].
Then:
(i)
We
have
Σ
α
=
Σ
β
.
(ii)
For
m
∈
{1,
.
.
.
,
n},
φ
m
induces
a
bijection
between
the
set
of
divisors
α
of
Π
m
and
the
set
of
divisors
of
Π
βm
.
β
(iii)
For
m
∈
{1,
.
.
.
,
n},
suppose
that
c
α
,
c
β
are
divisors
of
Π
α
m
,
Π
m
,
respec-
tively,
that
correspond
via
the
bijection
of
(ii).
Then
φ
m
(I
c
α
)
=
I
c
β
,
φ
m
(D
c
α
)
=
D
c
β
.
That
is
to
say,
φ
m
is
compatible
with
the
inertia
and
decomposition
groups
of
divisors.
(iv)
For
m
∈
{0,
.
.
.
,
n
−
1},
the
isomorphism
∼
β
α
β
Ker(Π
α
m+1
Π
m
)
→
Ker(Π
m+1
Π
m
)
induced
by
φ
m+1
is
graphic
[i.e.,
compatible
with
the
semi-graphs
of
anabelioids
that
appear
in
the
respective
collections
of
construction
data
of
the
PSC-extensions
Π
m+1
Π
m
,
for
=
α,
β].
α
(v)
For
m
∈
{1,
.
.
.
,
n
−
1},
c
α
,
c
β
corresponding
divisors
of
Π
α
m
,
Π
m
,
the
isomorphism
∼
β
Ker((Π
α
m+1
)
c
α
D
c
α
)
→
Ker((Π
m+1
)
c
β
D
c
β
)
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
25
induced
by
φ
m+1
is
graphic
[i.e.,
compatible
with
the
semi-graphs
of
anabelioids
that
appear
in
the
respective
collections
of
construction
data
of
the
DPSC-extensions
(Π
m+1
)
c
D
c
,
for
=
α,
β].
Proof.
All
of
the
assertions
of
Theorem
1.7
follow
immediately
from
[the
various
definitions
involved,
together
with]
repeated
application
of
Corollary
1.4
to
the
PSC-
extensions
Π
m+1
Π
m
[cf.
Definition
1.5,
(ii),
(c)]
and
the
DPSC-extensions
(Π
m+1
)
c
D
c
[cf.
Definition
1.5,
(ii),
(d)],
for
=
α,
β.
Remark
1.7.1.
In
Theorem
1.7,
instead
of
phrasing
the
result
as
an
asser-
tion
concerning
the
preservation
of
structures
via
some
isomorphism
between
two
PPSC-extensions,
one
may
instead
phrase
the
result
as
an
assertion
concerning
the
existence
of
an
explicit
“group-theoretic
algorithm”
for
reconstructing,
from
a
single
given
PPSC-extension,
the
various
structures
corresponding
to
graphicity,
divisors,
and
inertia
and
decomposition
groups
of
divisors
—
i.e.,
in
the
fashion
of
[Mzk15],
Lemma
4.5,
for
cuspidal
decomposition
groups;
a
similar
remark
may
be
made
concerning
Corollary
1.4.
[We
leave
the
routine
details
to
the
interested
reader.]
Indeed,
both
Corollary
1.4
and
Theorem
1.7
are,
in
essence,
formal
consequences
of
the
“graphicity
theory”
of
[Mzk13],
which
[just
as
in
the
case
of
[Mzk15],
Lemma
4.5]
consists
precisely
of
such
explicit
“group-theoretic
algorithms”
for
reconstruct-
ing
the
various
structures
corresponding
to
graphicity
in
the
case
of
semi-graphs
of
anabelioids
of
PSC-type.
Before
proceeding,
we
observe
the
following
result,
which
is,
in
essence,
inde-
pendent
of
the
theory
of
the
present
§1.
Theorem
1.8.
(PPSC-Extensions
over
Galois
Groups
of
Arithmetic
k
a
solvably
Fields)
For
=
α,
β,
let
k
be
a
field
of
characteristic
zero;
def
k
/k
);
closed
[cf.
[Mzk15],
Definition
1.4]
Galois
extension
of
k
;
H
=
Gal(
(Z
)
log
the
log
scheme
determined
by
a
stable
polycurve
over
k
;
Σ
a
nonempty
set
of
primes;
n
a
positive
integer;
1
→
Δ
→
Π
H
→
1
a
pro-Σ
PPSC-
k
[cf.
Remark
1.5.4];
extension
of
dimension
n
associated
to
(Z
)
log
,
Π
n
=
Π
Π
n−1
.
.
.
Π
1
Π
0
=
H
def
def
the
sequence
of
successive
fiber
extensions
associated
to
Π
[cf.
Remark
1.5.1].
Let
∼
φ
:
Π
α
→
Π
β
∼
β
be
an
isomorphism
of
profinite
groups
that
induces
isomorphisms
φ
m
:
Π
α
m
→
Π
m
,
for
m
=
0,
1,
.
.
.
,
n
[so
φ
=
φ
n
].
Then:
(i)
(Relative
Version
of
the
Grothendieck
Conjecture
for
Stable
Poly-
curves
over
Generalized
Sub-p-adic
Fields)
Suppose
that
for
=
α,
β,
k
is
generalized
sub-p-adic
[cf.
[Mzk4],
Definition
4.11]
for
some
prime
number
∼
p
∈
Σ
α
Σ
β
,
and
that
the
isomorphism
of
Galois
groups
φ
0
:
H
α
→
H
β
arises
∼
∼
from
a
pair
of
isomorphisms
of
fields
k
α
→
k
β
,
k
α
→
k
β
.
Then
Σ
α
=
Σ
β
;
there
26
SHINICHI
MOCHIZUKI
∼
exists
a
unique
isomorphism
of
log
schemes
(Z
α
)
log
→
(Z
β
)
log
that
gives
rise
to
φ.
(ii)
(Absolute
Version
of
the
Grothendieck
Conjecture
for
Stable
Polycurves
over
Number
Fields)
Suppose
that
for
=
α,
β,
k
is
a
number
field.
Then
Σ
α
=
Σ
β
;
there
exists
a
unique
isomorphism
of
log
schemes
∼
(Z
α
)
log
→
(Z
β
)
log
that
gives
rise
to
φ.
Proof.
Assertions
(i),
(ii)
follow
immediately
from
repeated
application
of
[Mzk4],
Theorem
4.12
[cf.
also
[Mzk2],
Corollary
7.4],
together
with
[in
the
case
of
assertion
(ii)]
“Uchida’s
theorem”
[cf.,
e.g.,
[Mzk10],
Theorem
3.1].
Finally,
we
study
the
consequences
of
the
theory
of
the
present
§1
in
the
case
of
configuration
spaces.
We
refer
to
[MT]
for
more
details
on
the
theory
of
config-
uration
spaces.
Definition
1.9.
Let
l
be
a
prime
number;
Σ
a
set
of
primes
which
is
either
of
cardinality
one
or
equal
to
the
set
of
all
primes;
X
a
hyperbolic
curve
of
type
(g,
r)
over
a
field
k
of
characteristic
∈
Σ;
k
a
separable
closure
of
k;
n
≥
1
an
integer;
X
n
the
n-th
configuration
space
associated
to
X
[cf.
[MT],
Definition
2.1,
(i)];
E
the
index
set
[i.e.,
the
set
of
factors
—
cf.
[MT],
Definition
2.1,
(i)]
of
X
n
;
π
1
(X
n
×
k
k)
Θ
the
maximal
pro-Σ
quotient
of
π
1
(X
n
×
k
k);
Δ
⊆
Θ
a
product-theoretic
open
subgroup
[cf.
[MT],
Definition
2.3,
(ii)];
1
→
Δ
→
Π
→
H
→
1
an
extension
of
profinite
groups.
∼
(i)
We
shall
refer
to
as
a
labeling
on
E
a
bijection
Λ
:
{1,
2,
.
.
.
,
n}
→
E.
Thus,
for
each
labeling
Λ
on
E,
we
obtain
a
structure
of
hyperbolic
polycurve
[i.e.,
a
collection
of
data
exhibiting
X
n
as
a
hyperbolic
polycurve
—
cf.
[Mzk2],
Definition
4.6]
on
X
n
,
arising
from
the
various
natural
projection
morphisms
associated
to
X
n
[cf.
[MT],
Definition
2.1,
(ii)],
by
projecting
in
the
order
specified
by
Λ.
In
particular,
for
each
labeling
Λ
on
E,
we
obtain
a
structure
of
PPSC-extension
on
[the
extension
1
→
Δ
Λ
→
Δ
Λ
→
{1}
→
1
associated
to]
some
open
subgroup
Δ
Λ
⊆
Δ
[which
may
be
taken
to
be
arbitrarily
small
—
cf.
Remark
1.5.2].
(ii)
Let
Λ
be
a
labeling
on
E.
Then
we
shall
refer
to
a
structure
of
PPSC-
extension
on
[the
extension
1
→
Δ
Λ
→
Π
Λ
→
H
Λ
→
1
arising
from
1
→
Δ
→
Π
→
H
→
1
by
intersecting
with]
an
open
subgroup
Π
Λ
⊆
Π
as
Λ-admissible
if
it
induces
the
structure
of
PPSC-extension
on
Δ
Λ
discussed
in
(i).
(iii)
We
shall
refer
to
as
a
structure
of
[pro-Σ]
CPSC-extension
[of
type
(g,
r)
and
dimension
n,
with
index
set
E]
[i.e.,
“configuration
(space)
pointed
stable
curve
extension”]
on
the
extension
1
→
Δ
→
Π
→
H
→
1
any
collection
of
data
as
follows:
for
each
labeling
Λ
on
E,
a
Λ-admissible
structure
of
PPSC-extension
on
some
open
subgroup
Π
Λ
⊆
Π
[which
may
be
taken
to
be
arbitrarily
small
—
cf.
Remark
1.5.2].
We
shall
refer
to
a
structure
of
CPSC-extension
on
Π
as
l-cyclotomically
full
if,
for
each
labeling
Λ
on
E,
the
Λ-admissible
structure
of
PPSC-extension
that
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
27
constitutes
the
given
structure
of
CPSC-extension
is
l-cyclotomically
full.
We
shall
refer
to
a
structure
of
CPSC-extension
on
Π
as
strict
if
one
may
take
Δ
to
be
Θ,
and,
for
each
labeling
Λ
on
E,
one
may
take
Π
Λ
to
be
Π.
We
shall
refer
to
1
→
Δ
→
Π
→
H
→
1
as
a
[pro-Σ]
CPSC-extension
[of
type
(g,
r)
and
dimension
n,
with
index
set
E]
if
it
admits
a
structure
of
CPSC-extension;
if
this
structure
of
CPSC-extension
may
be
taken
to
be
l-cyclotomically
full
(respectively,
strict),
then
we
shall
refer
to
the
CPSC-extension
itself
as
l-cyclotomically
full
(respectively,
strict).
If
1
→
Δ
→
Π
→
H
→
1
is
a
CPSC-extension,
then
we
shall
refer
to
(Σ,
X,
k,
Θ)
as
construction
data
for
this
CPSC-extension.
(iv)
Let
k
⊆
k
be
a
solvably
closed
[cf.
[Mzk15],
Definition
1.4]
Galois
extension
of
k;
suppose
that
Z
→
X
n
is
a
finite
étale
covering
such
that
Z
×
k
k
→
X
n
×
k
k
is
the
[connected]
covering
determined
by
the
open
subgroup
Δ
⊆
Θ
[so
we
have
a
natural
surjection
π
1
(Z
×
k
k)
Δ].
Then
[cf.
the
discussion
of
Remark
1.5.4]
we
shall
refer
to
a
[structure
of]
CPSC-extension
[on]
1
→
Δ
→
Π
→
H
→
1
as
arising
from
Z,
k/k
if
there
exist
∼
a
surjection
π
1
(Z)
Π
and
an
isomorphism
Gal(
k/k)
→
H
that
are
compatible
with
one
another
as
well
as
with
the
natural
surjections
π
1
(Z
×
k
k)
Δ,
π
1
(Z)
Gal(k/k)
Gal(
k/k)
and,
moreover,
satisfy
the
property
that
the
structure
of
CPSC-extension
on
1
→
Δ
→
Π
→
H
→
1
is
induced
by
the
various
structures
of
hyperbolic
polycurve
on
Z,
X
n
,
associated
to
a
suitable
labeling
of
E
[cf.
(i)].
Corollary
1.10.
(Cominatorial
Configuration
Spaces)
Let
l
be
a
prime
number.
For
=
α,
β,
let
1
→
Δ
→
Π
→
H
→
1
be
an
extension
of
profinite
groups
equipped
with
some
[fixed!]
l-cyclotomically
full
structure
of
CPSC-extension
of
type
(g
,
r
)
∈
/
{(0,
3),
(1,
1)}
and
dimen-
sion
n
,
with
index
set
E
.
If
this
fixed
structure
of
CPSC-extension
is
not
strict
for
either
=
α
or
=
β,
then
we
assume
that
both
g
α
,
g
β
are
≥
2.
Let
∼
φ
:
Π
α
→
Π
β
be
an
isomorphism
of
profinite
groups
such
that
φ(Δ
α
)
=
Δ
β
.
Then:
∼
(i)
The
isomorphism
φ
determines
a
bijection
E
α
→
E
β
of
index
sets.
In
def
particular,
n
α
=
n
β
,
so
we
write
n
=
n
α
=
n
β
.
(ii)
For
each
pair
of
compatible
[i.e.,
relative
to
the
bijection
of
(i)]
labelings
Λ
=
(Λ
α
,
Λ
β
)
of
E
α
,
E
β
,
there
exist
open
subgroups
Π
[for
=
α,
β]
Λ
⊆
Π
β
α
such
that
the
following
properties
hold:
(a)
φ(Π
Λ
)
=
Π
Λ
;
(b)
for
=
α,
β,
the
open
subgroup
Π
Λ
admits
an
Λ
-admissible
structure
of
PPSC-extension;
(c)
if
we
write
(Π
Λ
)
n
=
(Π
Λ
)
(Π
Λ
)
n−1
.
.
.
(Π
Λ
)
1
(Π
Λ
)
0
=
H
Λ
def
def
28
SHINICHI
MOCHIZUKI
for
the
sequence
of
successive
fiber
extensions
associated
to
the
structures
of
PPSC-extension
of
(b)
[cf.
Remark
1.5.1],
then
φ
induces
isomorphisms
∼
β
(Π
α
Λ
)
m
→
(Π
Λ
)
m
[for
m
=
0,
.
.
.
,
n].
In
particular,
φ
satisfies
the
hypotheses
of
Theorem
1.7.
Proof.
By
[MT],
Corollaries
4.8,
6.3
[cf.
our
hypotheses
on
(g
,
r
)],
φ
induces
∼
a
bijection
E
α
→
E
β
between
the
respective
index
sets,
together
with
compatible
isomorphisms
between
the
various
fiber
subgroups
of
Δ
α
,
Δ
β
.
[Note
that
even
though
these
results
of
[MT]
are
stated
only
in
the
case
where
the
field
appearing
in
the
construction
data
is
of
characteristic
zero,
the
results
generalize
immediately
to
the
case
where
this
field
is
of
characteristic
invertible
in
Σ
,
since
any
hyperbolic
curve
in
positive
characteristic
may
be
lifted
to
a
hyperbolic
curve
in
characteristic
zero
in
a
fashion
that
is
compatible
with
the
maximal
pro-Σ
quotients
of
the
étale
fundamental
groups
of
the
associated
configuration
spaces
—
cf.,
e.g.,
[MT],
Proposition
2.2,
(v).]
To
obtain
open
subgroups
Π
satisfying
the
desired
Λ
⊆
Π
properties,
it
suffices
to
argue
by
induction
on
n,
by
applying
Remark
1.5.2.
Remark
1.10.1.
1.10.
A
similar
remark
to
Remark
1.7.1
may
be
made
for
Corollary
Corollary
1.11.
(Configuration
Spaces
over
Arithmetic
Fields)
For
=
k
a
solvably
closed
[cf.
[Mzk15],
Definition
1.4]
α,
β,
let
k
be
a
perfect
field;
/
{(0,
3),
(1,
1)}
Galois
extension
of
k
;
X
a
hyperbolic
curve
of
type
(g
,
r
)
∈
over
k
;
n
a
positive
integer;
Z
→
(X
)
n
a
geometrically
connected
[over
k
]
finite
étale
covering
of
the
n
-th
configu-
ration
space
(X
)
n
of
X
;
Σ
a
nonempty
set
of
primes;
1
→
Δ
→
Π
→
H
→
1
an
extension
of
profinite
groups
equipped
with
some
[fixed!]
structure
of
k
[cf.
Definition
1.9,
(iv)].
If
this
pro-Σ
CPSC-extension
arising
from
Z
,
fixed
structure
of
CPSC-extension
is
not
strict
for
either
=
α
or
=
β,
then
we
assume
that
both
g
α
,
g
β
are
≥
2.
Let
∼
φ
:
Π
α
→
Π
β
be
an
isomorphism
of
profinite
groups.
Then:
(i)
(Relative
Version
of
the
Grothendieck
Conjecture
for
Configura-
tion
Spaces
over
Generalized
Sub-p-adic
Fields)
Suppose,
for
=
α,
β,
that
k
is
generalized
sub-p-adic
[cf.
[Mzk4],
Definition
4.11]
for
some
prime
number
∼
p
∈
Σ
α
Σ
β
,
and
that
φ
lies
over
an
isomorphism
of
Galois
groups
φ
0
:
H
α
→
H
β
∼
∼
that
arises
from
a
pair
of
isomorphisms
of
fields
k
α
→
k
β
,
k
α
→
k
β
.
Then
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
29
∼
Σ
α
=
Σ
β
;
there
exists
a
unique
isomorphism
of
schemes
Z
α
→
Z
β
that
gives
rise
to
φ.
(ii)
(Strict
Semi-absoluteness)
Suppose,
for
=
α,
β,
that
k
is
either
an
FF,
an
MLF,
or
an
NF
[cf.
[Mzk15],
§0].
Then
φ(Δ
α
)
=
Δ
β
[i.e.,
φ
is
“strictly
semi-absolute”].
(iii)
(Absolute
Version
of
the
Grothendieck
Conjecture
for
Configu-
ration
Spaces
over
MLF’s)
Suppose,
for
=
α,
β,
that
k
is
an
MLF,
that
n
≥
2,
that
n
≥
3
if
X
is
proper,
and
that
Σ
is
the
set
of
all
primes.
Then
∼
there
exists
a
unique
isomorphism
of
schemes
Z
α
→
Z
β
that
gives
rise
to
φ.
(iv)
(Absolute
Version
of
the
Grothendieck
Conjecture
for
Config-
uration
Spaces
over
NF’s)
Suppose,
for
=
α,
β,
that
k
is
an
NF.
Then
∼
Σ
α
=
Σ
β
;
there
exists
a
unique
isomorphism
of
schemes
Z
α
→
Z
β
that
gives
rise
to
φ.
Proof.
Assertion
(i)
(respectively,
(iv))
follows
immediately
from
Corollary
1.10,
(ii),
and
Theorem
1.8,
(i)
(respectively,
Theorem
1.8,
(ii))
[applied
to
the
coverings
of
Z
α
,
Z
β
determined
by
the
open
subgroups
of
Corollary
1.10,
(ii)].
Assertion
(ii)
follows
immediately
from
[Mzk15],
Corollary
2.8,
(ii).
Note
that
in
the
situation
of
assertion
(ii),
assertion
(ii)
implies
that
Σ
α
=
Σ
β
[since
Σ
may
be
characterized
as
the
unique
minimal
set
of
primes
Σ
such
that
Δ
is
a
pro-Σ
group];
moreover,
in
light
of
our
assumptions
β
on
α
k
,
it
β
follows
immediately
that
Π
is
l-cyclotomically
α
Σ
=Σ
=Σ
.
full
for
any
l
∈
Σ
Finally,
we
consider
assertion
(iii).
First,
let
us
observe
that
by
Corollary
1.10,
(ii),
and
Theorem
1.8,
(i)
[applied
to
the
coverings
of
Z
α
,
Z
β
determined
by
the
open
subgroups
of
Corollary
1.10,
(ii)],
it
suffices
to
verify
that
the
isomorphism
∼
φ
H
:
H
α
→
H
β
induced
by
φ
[cf.
assertion
(ii)]
arises
from
an
isomorphism
of
fields
∼
k
α
→
k
β
.
To
this
end,
let
us
observe
that
by
Corollary
1.10,
(ii),
we
may
apply
Theorem
1.7
to
the
present
situation.
Also,
by
Corollary
1.10,
(i),
n
=
n
α
=
n
β
is
always
≥
2;
moreover,
if
either
of
the
X
is
proper,
then
n
≥
3.
Next,
let
us
observe
that
if
X
is
proper
(respectively,
affine),
then
the
stable
log
curve
that
appears
in
the
logarithmic
compactification
of
the
fibration
(X
)
3
→
(X
)
2
(respectively,
(X
)
2
→
(X
)
1
)
over
the
generic
point
of
the
diagonal
divisor
of
(X
)
2
(respectively,
over
any
cusp
of
X
)
contains
an
irreducible
component
whose
interior
is
a
tripod
[i.e.,
a
copy
of
the
projective
line
minus
three
marked
points].
In
particular,
if
we
apply
Theorem
1.7,
(iii),
to
the
vertical
divisor
determined
by
such
an
irreducible
component,
then
we
may
conclude
that
φ
induces
an
isomorphism
between
the
decomposition
groups
of
these
vertical
divisors.
In
particular,
[after
possibly
replacing
the
given
k
α
,
k
β
by
corresponding
finite
extensions
of
k
α
,
k
β
]
we
obtain,
for
=
α,
β,
a
hyperbolic
curve
C
over
k
,
together
with
an
isomorphism
of
profinite
groups
∼
k
α
)
→
π
1
(C
β
×
k
β
k
β
)
φ
C
:
π
1
(C
α
×
k
α
k
)
correspond
to
the
respective
“Π
v
’s”
of
the
induced
by
φ
[so
the
π
1
(C
×
k
vertical
divisors
under
consideration]
that
is
compatible
with
the
outer
action
of
30
SHINICHI
MOCHIZUKI
k
)
and
the
isomorphism
φ
H
;
moreover,
here
we
may
assume
H
on
π
1
(C
×
k
that,
say,
C
α
is
a
finite
étale
covering
of
a
tripod.
[In
particular,
we
observe
that
k
)
implies
—
cf.
the
the
existence
of
the
natural
outer
action
of
H
on
π
1
(C
×
k
argument
given
in
the
proof
of
[Mzk10],
Proposition
3.3
—
that
k
is
necessarily
an
algebraic
closure
of
k
.]
On
the
other
hand,
since
the
“absolute
p-adic
version
of
the
Grothendieck
Conjecture”
is
known
to
hold
in
this
situation
[cf.
[Mzk14],
Corollary
2.3],
we
thus
conclude
that
φ
H
does
indeed
arise
from
an
isomorphism
of
∼
fields
k
α
→
k
β
,
as
desired.
This
completes
the
proof
of
assertion
(iii).
Remark
1.11.1.
At
the
time
of
writing
Corollary
1.11,
(iii),
constitutes
the
only
absolute
isomorphism
version
of
the
Grothendieck
Conjecture
over
MLF’s
[to
the
knowledge
of
the
author]
that
may
be
applied
to
arbitrary
hyperbolic
curves.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
31
Section
2:
Geometric
Uniformly
Toral
Neighborhoods
In
the
present
§2,
we
prove
a
certain
“resolution
of
nonsingularities”
type
re-
sult
[cf.
Lemma
2.6;
Remark
2.6.1;
Corollary
2.11]
—
i.e.,
a
result
reminiscent
of
the
main
results
of
[Tama2]
[cf.
also
the
techniques
applied
in
the
verification
of
“observation
(iv)”
given
in
the
proof
of
[Mzk9],
Corollary
3.11]
—
that
allows
us
to
apply
the
theory
of
uniformly
toral
neighborhoods
developed
in
[Mzk15],
§3,
to
prove
a
certain
“conditional
absolute
p-adic
version
of
the
Grothendieck
Conjec-
ture”
—
namely,
that
“point-theoreticity
implies
geometricity”
[cf.
Corollary
2.9].
This
condition
of
point-theoreticity
may
be
removed
if,
instead
of
starting
with
a
hyperbolic
curve,
one
starts
with
a
“pro-curve”
obtained
by
removing
from
a
proper
curve
some
[necessarily
infinite]
set
of
closed
points
which
is
“p-adically
dense
in
a
Galois-compatible
fashion”
[cf.
Corollary
2.10].
First,
we
recall
the
following
“positive
slope
version
of
Hensel’s
lemma”
[cf.
[Serre],
Chapter
II,
§2.2,
Theorem
1,
for
a
discussion
of
a
similar
result].
Lemma
2.1.
(Positive
Slope
Version
of
Hensel’s
Lemma)
Let
k
be
a
complete
discretely
valued
field;
O
k
⊆
k
the
ring
of
integers
of
k
[equipped
with
the
topology
determined
by
the
valuation];
m
k
the
maximal
ideal
of
O
k
;
π
∈
m
k
a
def
def
uniformizer
of
O
k
;
A
=
O
k
[[X
1
,
.
.
.
,
X
m
]];
B
=
O
k
[[Y
1
,
.
.
.
,
Y
n
]].
Let
us
suppose
that
A
(respectively,
B)
is
equipped
with
the
topology
determined
by
its
maximal
def
def
ideal;
write
X
=
Spf(A)
(respectively,
Y
=
Spf(B)),
K
A
(respectively,
K
B
)
for
the
quotient
field
of
A
(respectively,
B),
and
Ω
A
(respectively,
Ω
B
)
for
the
module
of
continuous
differentials
of
A
(respectively,
B)
over
O
k
[so
Ω
A
(respectively,
Ω
B
)
is
a
free
A-
(respectively,
B-)
module
of
rank
m
(respectively,
n)].
Let
φ
:
B
→
A
be
the
continuous
O
k
-algebra
homomorphism
induced
by
an
assignment
B
Y
j
→
f
j
(X
1
,
.
.
.
,
X
m
)
∈
A
[where
j
=
1,
.
.
.
,
n];
let
us
suppose
that
the
induced
morphism
dφ
:
Ω
B
⊗
B
A
→
Ω
A
satisfies
the
property
that
the
image
of
dφ
⊗
A
K
A
is
a
K
A
-subspace
of
rank
n
in
Ω
A
⊗
A
K
A
[so
n
≤
m].
Then
there
exists
a
point
β
0
∈
Y(O
k
)
and
a
positive
integer
r
satisfying
the
following
property:
Let
k
be
a
finite
extension
of
k,
with
ring
of
integers
O
k
;
write
B(β
0
,
k
,
r)
for
the
“ball”
of
points
β
∈
Y(O
k
)
such
that
β
,
β
0
map
to
the
same
point
of
Y(O
k
/(π
r
)).
Then
the
image
of
the
map
X
(O
k
)
→
Y(O
k
)
induced
by
φ
contains
the
“ball”
B(β
0
,
k
,
r).
Proof.
First,
let
us
observe
that
by
Lemma
2.2
below,
after
possibly
re-ordering
the
X
i
’s,
we
may
assume
that
the
differentials
dX
i
∈
Ω
A
,
where
i
=
n
+
1,
.
.
.
,
m,
together
with
the
differentials
df
j
∈
Ω
A
,
where
j
=
1,
.
.
.
,
n,
form
a
K
A
-basis
of
Ω
A
⊗
A
K
A
.
Thus,
by
adding
indeterminates
Y
n+1
,
.
.
.
,
Y
m
to
B
and
extending
φ
by
sending
Y
i
→
X
i
for
i
=
n
+
1,
.
.
.
,
m,
we
may
assume
without
loss
of
generality
that
n
=
m,
A
=
B,
X
=
Y,
i.e.,
that
the
morphism
Spf(φ)
:
X
→
Y
=
X
is
“generically
formally
étale”.
32
SHINICHI
MOCHIZUKI
Write
M
for
the
n
by
n
matrix
with
coefficients
∈
A
given
by
{df
i
/dX
j
}
i,j=1,...
,n
;
g
∈
A
for
the
determinant
of
M
.
Thus,
by
elementary
linear
algebra,
it
follows
that
there
exists
an
n
by
n
matrix
N
with
coefficients
∈
A
such
that
M
·N
=
N
·M
=
g·I
[where
we
write
I
for
the
n
by
n
identity
matrix].
By
our
assumption
concerning
the
image
of
dφ⊗
A
K
A
,
it
follows
that
g
=
0,
hence,
by
Lemma
2.3
below,
that
there
def
exist
elements
x
i
∈
m
k
,
where
i
=
1,
.
.
.
,
n,
such
that
g
0
=
g(x
1
,
.
.
.
,
x
n
)
∈
m
k
is
nonzero.
By
applying
appropriate
“affine
translations”
to
the
domain
and
codomain
of
φ,
we
may
assume
without
loss
of
generality
that,
for
i
=
1,
.
.
.
,
n,
we
have
def
def
x
i
=
f
i
(0,
.
.
.
,
0)
=
0
∈
O
k
.
Write
M
0
=
M
(0,
.
.
.
,
0),
N
0
=
N
(0,
.
.
.
,
0)
[so
M
0
,
N
0
are
n
by
n
matrices
with
coefficients
∈
O
k
].
Next,
suppose
that
g
0
∈
m
sk
\m
s+1
k
.
In
the
remainder
of
the
present
proof,
all
“vectors”
are
to
be
understood
as
column
vectors
with
coefficients
∈
O
k
,
where
k
is
as
in
the
statement
of
Lemma
2.1.
Then
I
claim
that
for
every
vector
y
=
(y
1
,
.
.
.
,
y
n
)
≡
0
(mod
π
3s
),
there
exists
a
vector
x
=
(x
1
,
.
.
.
,
x
n
)
≡
0
(mod
π
2s
)
def
such
that
f
(x
)
=
(f
1
(x
),
.
.
.
,
f
n
(x
))
=
y
.
Indeed,
since
O
k
is
complete
and
x
i
=
f
i
(0,
.
.
.
,
0)
=
0,
it
suffices
[by
taking
x
{2}
to
be
(0,
.
.
.
,
0)]
to
show,
for
each
integer
l
≥
2,
that
the
existence
of
a
vector
x
{l}
≡
0
(mod
π
2s
)
such
that
f
(x
{l})
≡
y
(mod
π
(l+1)s
)
implies
the
existence
of
a
vector
x
{l
+
1}
such
that
x
{l
+
1}
≡
x
{l}
def
(mod
π
ls
),
f
(x
{l
+
1})
≡
y
(mod
π
(l+2)s
).
To
this
end,
we
compute:
Set
=
def
def
def
y
−
f
(x
{l}),
η
=
g
0
−1
·
,
δ
=
N
0
·η
,
x
{l
+
1}
=
x
{l}+
δ.
Thus,
≡
0
(mod
π
(l+1)s
),
η
≡
0
(mod
π
ls
),
δ
≡
0
(mod
π
ls
),
x
{l
+
1}
≡
x
{l}
(mod
π
ls
).
In
particular,
the
squares
of
the
elements
of
δ
all
belong
to
π
(l+2)s
·
O
k
[since
l
≥
2],
so
we
obtain
that
f
(x
{l
+
1})
≡
f
(x
{l}
+
δ)
≡
f
(x
{l})
+
M
0
·
δ
(mod
π
(l+2)s
)
≡
y
−
+
M
0
·
N
0
·
η
≡
y
−
+
g
0
·
η
≡
y
(mod
π
(l+2)s
)
—
as
desired.
This
completes
the
proof
of
the
claim.
On
the
other
hand,
one
verifies
immediately
that
the
content
of
this
claim
is
sufficient
to
complete
the
proof
of
Lemma
2.1.
Remark
2.1.1.
Thus,
the
usual
“(slope
zero
version
of
)
Hensel’s
lemma”
corre-
sponds,
in
the
notation
of
Lemma
2.1,
to
the
case
where
the
image
of
the
morphism
dφ
is
a
direct
summand
of
Ω
A
.
In
this
case,
we
may
take
r
=
1.
Remark
2.1.2.
In
fact,
according
to
oral
communication
to
the
author
by
F.
Oort,
it
appears
that
the
sort
of
“positive
slope
version”
of
“Hensel’s
lemma”
given
in
Lemma
2.1
[i.e.,
where
the
derivative
is
only
generically
invertible]
preceded
the
“slope
zero
version”
that
is
typically
referred
to
“Hensel’s
lemma”
in
modern
treatments
of
the
subject.
Lemma
2.2.
(Subspaces
and
Bases
of
a
Vector
Space)
Let
k
be
a
field;
V
a
finite-dimensional
k-vector
space
with
basis
{e
i
}
i∈I
;
W
⊆
V
a
k-subspace.
Then
there
exists
a
subset
J
⊆
I
such
that
if
we
write
V
J
⊆
V
for
the
k-subspace
generated
by
the
e
j
,
for
j
∈
J,
then
the
natural
inclusions
V
J
→
V
,
W
→
V
determine
an
∼
isomorphism
V
J
⊕
W
→
V
of
k-vector
spaces.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
33
Proof.
This
result
is
a
matter
of
elementary
linear
algebra.
Lemma
2.3.
(Nonzero
Values
of
Functions
Defined
by
Power
Series)
Let
k,
O
k
,
A
be
as
in
Lemma
2.1;
f
=
f
(X
1
,
.
.
.
,
X
m
)
∈
A
a
nonzero
element.
Then
there
exist
elements
x
i
∈
m
k
,
where
i
=
1,
.
.
.
,
m,
such
that
f
(x
1
,
.
.
.
,
x
m
)
∈
m
k
is
nonzero.
Proof.
First,
I
claim
that
by
induction
on
m,
it
suffices
to
verify
Lemma
2.3
when
m
=
1.
Indeed,
for
arbitrary
m
≥
2,
one
may
write
f
=
∞
i
c
i
X
m
i=0
—
where
c
i
=
c
i
(X
1
,
.
.
.
,
X
m−1
)
∈
O
k
[[X
1
,
.
.
.
,
X
m−1
]].
Since
f
=
0,
it
follows
that
there
exists
at
least
one
nonzero
c
j
.
Thus,
by
the
induction
hypothesis,
it
follows
that
there
exist
x
i
∈
m
k
,
where
i
=
1,
.
.
.
,
m−1,
such
that
m
k
c
j
(x
1
,
.
.
.
,
x
m−1
)
=
0.
Thus,
f
(x
1
,
.
.
.
,
x
m−1
,
X
m
)
∈
O
k
[[X
m
]]
is
nonzero,
so,
again
by
the
induction
hypothesis,
there
exists
an
x
m
∈
m
k
such
that
m
k
f
(x
1
,
.
.
.
,
x
m−1
,
x
m
)
=
0.
This
completes
the
proof
of
the
claim.
def
Thus,
for
the
remainder
of
the
proof,
we
assume
that
m
=
1
and
write
X
=
X
1
,
f
=
∞
c
i
X
i
i=0
—
where
c
i
∈
O
k
.
Suppose
that
c
j
=
0,
but
that
c
i
=
0
for
i
<
j.
Then
there
exists
a
positive
integer
s
such
that
c
j
∈
m
sk
.
Let
x
∈
m
sk
be
any
nonzero
element.
Then
c
j
x
j
∈
x
j
·
m
sk
,
while
c
i
x
i
∈
x
j
·
x
·
O
k
⊆
x
j
·
m
sk
for
any
i
>
j.
But
this
implies
that
m
k
f
(x)
=
0,
as
desired.
In
the
following,
we
shall
often
work
with
[two-dimensional]
log
regular
log
schemes.
For
various
basic
facts
on
log
regular
log
schemes,
we
refer
to
[Kato];
[Mzk2],
§1.
If
X
log
is
a
log
regular
log
scheme,
then
for
integers
j
≥
0,
we
shall
write
[j]
U
X
⊆
X
for
the
j-interior
of
X
log
,
i.e.,
the
open
subscheme
of
points
at
which
the
fiber
of
the
groupification
of
the
characteristic
sheaf
of
X
log
is
of
rank
≤
j
[cf.
[MT],
[j]
Definition
5.1,
(i);
[MT],
Proposition
5.2,
(i)].
Thus,
the
complement
of
U
X
in
X
def
[0]
is
a
closed
subset
of
codimension
>
j
[cf.
[MT],
Proposition
5.2,
(ii)];
U
X
=
U
X
is
the
interior
of
X
log
[i.e.,
the
maximal
open
subscheme
over
which
the
log
structure
is
trivial].
Also,
we
shall
write
D
X
⊆
X
for
the
closed
subscheme
X\U
X
with
the
reduced
induced
structure.
Finally,
we
remind
the
reader
that
in
the
following,
all
fiber
products
of
fs
log
schemes
are
to
be
taken
in
the
category
of
fs
log
schemes
[cf.
§0].
34
SHINICHI
MOCHIZUKI
Next,
let
k
be
a
complete
discretely
valued
field
with
perfect
residue
field
k;
O
k
⊆
k
the
ring
of
integers
of
k;
k
an
algebraic
closure
of
k;
k
the
resulting
algebraic
closure
of
k;
m
k
the
maximal
ideal
of
O
k
;
π
∈
m
k
a
uniformizer
of
O
k
;
def
X
→
S
=
Spec(O
k
)
def
def
a
stable
curve
over
S
whose
generic
fiber
X
η
=
X
×
S
η,
where
we
write
η
=
Spec(k),
is
smooth.
Thus,
X
η
is
a
proper
hyperbolic
curve
over
k,
whose
genus
we
denote
by
g
X
;
the
open
subschemes
η
⊆
S,
X
η
⊆
X
determine
log
regular
log
structures
on
S,
X,
respectively.
We
denote
the
resulting
morphism
of
log
schemes
by
X
log
→
S
log
.
Definition
2.4.
(i)
We
shall
refer
to
a
morphism
of
log
schemes
φ
log
:
V
log
→
X
log
[or
to
the
log
scheme
V
log
]
as
a
log-modification
if
φ
log
admits
a
factorization
V
log
→
X
log
×
S
log
S
V
log
→
X
log
def
—
where
S
V
=
Spec(O
k
V
),
O
k
V
is
the
ring
of
integers
of
a
finite
separable
extension
def
k
V
of
k;
S
V
log
is
the
log
regular
log
scheme
determined
by
the
open
immersion
η
V
=
Spec(k
V
)
→
S
V
;
the
morphism
X
log
×
S
log
S
V
log
→
X
log
is
the
projection
morphism
[where
we
observe
that
the
underlying
morphism
of
schemes
X
×
S
S
V
→
S
V
is
a
stable
curve
over
S
V
];
the
morphism
V
log
→
X
log
×
S
log
S
V
log
is
a
log
étale
morphism
whose
underlying
morphism
of
schemes
is
proper
and
birational;
we
shall
refer
to
k
V
as
the
base-field
of
the
log-modification
φ
log
.
(ii)
For
i
=
1,
2,
let
φ
log
:
V
i
log
→
X
log
be
a
log-modification
that
admits
a
i
factorization
V
i
log
→
X
log
×
S
log
S
i
log
→
X
log
as
in
(i);
ψ
log
:
V
2
log
→
V
1
log
an
X
log
-
morphism.
Then
let
us
observe
that
the
log
scheme
V
i
log
is
always
log
regular
of
dimension
2
[cf.
Proposition
2.5,
(iv),
below].
We
shall
refer
to
the
log-modification
φ
log
as
regular
if
the
log
structure
of
V
i
log
is
defined
by
a
divisor
with
normal
i
crossings
[which
implies
that
V
i
is
a
regular
scheme].
We
shall
refer
to
the
log-
[1]
modification
φ
log
as
unramified
if
U
V
i
is
a
smooth
scheme
over
S
i
.
We
shall
refer
i
to
the
morphism
ψ
log
as
a
base-field-isomorphism
[or
base-field-isomorphic]
if
the
morphism
S
2
→
S
1
induced
by
ψ
log
is
an
isomorphism.
We
shall
refer
to
the
points
of
V
1
over
which
the
underlying
morphism
ψ
of
ψ
log
fails
to
be
finite
as
the
critical
points
of
ψ
log
[or
ψ].
We
shall
refer
to
the
reduced
divisor
in
V
2
determined
by
the
closed
set
of
points
of
V
2
at
which
ψ
fails
to
be
quasi-finite
as
the
exceptional
divisor
of
ψ
log
[or
ψ].
We
shall
refer
to
the
log
scheme-
(respectively,
scheme-)
theoretic
fiber
of
V
i
log
(respectively,
V
i
)
over
the
unique
closed
point
of
S
i
as
the
log
special
fiber
(respectively,
special
fiber)
of
V
i
log
(respectively,
V
i
);
we
shall
use
the
notation
V
log
(respectively,
V
i
)
i
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
35
to
denote
the
log
special
fiber
(respectively,
special
fiber)
of
V
i
log
(respectively,
V
i
).
Suppose
that
C
is
an
irreducible
component
of
V
i
.
Then
we
shall
say
that
C
is
stable
if
it
maps
finitely
to
X
via
φ
i
;
we
shall
say
that
the
log-modification
φ
log
as
i
[1]
unramified
at
C
if
U
V
i
is
smooth
over
S
i
at
the
generic
point
of
C.
Remark
2.4.1.
Recall
that
there
exists
a
base-field
isomorphic
log-modification
V
log
→
X
log
which
is
uniquely
determined
up
to
unique
isomorphism
[over
X]
by
the
follow-
ing
two
properties:
(a)
V
is
regular;
(b)
V
→
S
is
a
semi-stable
curve.
In
fact,
it
was
this
example
that
served
as
the
primary
motivating
example
for
the
au-
thor
in
developing
the
notion
of
a
“log-modification”.
Note,
moreover,
that
unlike
property
(a),
however,
the
principal
condition
that
defines
a
[base-field-isomorphic]
“log-modification”
—
i.e.,
the
condition
that
the
morphism
V
→
X
be
a
proper,
bi-
rational
morphism
that
extends
to
a
log
étale
morphism
V
log
→
X
log
of
log
schemes
—
is
a
condition
on
the
morphism
V
→
X
that
has
the
virtue
of
being
manifestly
stable
under
base-change
[i.e.,
via
morphisms
that
satisfy
suitable,
relatively
mild
conditions].
This
property
of
stability
under
base-change
will
be
applied
repeatedly
in
the
remainder
of
the
present
§2.
Proposition
2.5.
(First
Properties
of
Log-modifications)
For
i
=
1,
2,
let
:
V
i
log
→
X
log
φ
log
i
be
a
log-modification
that
admits
a
factorization
V
i
log
→
X
log
×
S
log
S
i
log
→
X
log
as
in
Definition
2.4,
(i);
ψ
log
:
V
2
log
→
V
1
log
an
X
log
-morphism.
Write
S
i
=
Spec(O
k
i
),
where,
for
simplicity,
we
assume
that
the
extension
k
i
of
k
is
a
subfield
of
k;
k
i
for
the
residue
field
of
k
i
;
U
ψ
noncr
⊆
V
1
for
the
open
subscheme
given
by
the
complement
of
the
critical
points
of
ψ
log
.
Let
•
∈
{1,
2}.
Then:
[1]
(i)
(The
Noncriticality
of
the
1-Interior)
We
have:
U
V
1
⊆
U
ψ
noncr
.
(ii)
(Isomorphism
over
the
Noncritical
Locus)
The
morphism
V
2
log
→
V
1
log
×
S
log
S
2
log
determined
by
ψ
log
is
an
isomorphism
over
U
ψ
noncr
.
1
(iii)
(Log
Smoothness
and
Unramified
Log-modifications)
V
•
log
is
log
smooth
over
S
•
log
.
In
particular,
the
sheaf
of
relative
logarithmic
differentials
of
the
morphism
V
•
log
→
S
•
log
is
a
line
bundle,
which
we
shall
denote
ω
V
log
/S
log
;
•
•
we
have
a
natural
isomorphism
ψ
∗
ω
V
log
/S
log
∼
=
ω
V
log
/S
log
.
Finally,
there
exists
a
1
1
2
2
def
finite
extension
k
◦
of
k
•
such
that,
if
we
write
S
◦
=
Spec(O
k
◦
),
then
the
mor-
def
phism
V
◦
log
=
V
•
log
×
S
log
S
◦
log
→
X
log
determined
by
φ
log
•
,
is
an
unramified
log-
•
modification.
[1]
(iv)
(Regularity
and
Log
Regularity)
V
•
log
is
log
regular;
U
V
•
is
regu-
lar.
Moreover,
there
exists
a
regular
log-modification
V
◦
log
→
X
log
that
admits
36
SHINICHI
MOCHIZUKI
a
base-field-isomorphic
X
log
-morphism
V
◦
log
→
V
•
log
such
that
every
irreducible
component
C
of
the
special
fiber
V
◦
is
smooth
over
the
residue
field
k
◦
of
the
base-field
k
◦
of
V
◦
log
.
Finally,
if
the
log-modification
V
◦
log
→
X
log
is
unramified
at
such
an
irreducible
component
C,
and
we
write
D
C
⊆
C
for
the
reduced
divisor
[1]
determined
by
the
complement
of
C
U
V
◦
in
C,
then
we
have
a
natural
isomor-
phism
∼
ω
C/k
◦
(D
C
)
→
ω
V
log
/S
log
|
C
◦
◦
of
line
bundles
on
C.
is
a
regular
log-
(v)
(Chains
of
Projective
Lines)
Suppose
that
φ
log
2
modification
[so
D
V
2
is
a
divisor
with
normal
crossings].
Then,
after
possibly
replacing
k
2
by
a
finite
unramified
extension
of
the
discretely
valued
field
k
2
,
ev-
ery
irreducible
component
C
of
D
V
2
that
lies
in
the
exceptional
divisor
of
ψ
log
is
isomorphic
to
the
projective
line
over
the
residue
field
k(c)
of
the
point
c
of
V
1
log
×
S
log
S
2
log
to
which
C
maps.
Moreover,
C
meets
the
other
irreducible
com-
ponents
of
D
V
2
at
precisely
two
k(c)-valued
points
of
C.
That
is
to
say,
every
connected
component
of
the
exceptional
divisor
of
ψ
log
is
a
“chain
of
P
1
’s”.
(vi)
(Dual
Graphs
of
Special
Fibers)
The
spectrum
V
x
of
the
local
ring
obtained
by
completing
the
geometric
special
fiber
V
•
×
k
•
k
at
any
point
x
which
does
not
(respectively,
does)
belong
to
the
1-interior
has
precisely
two
(respec-
tively,
precisely
one)
irreducible
component(s).
In
particular,
the
special
fiber
V
•
determines
a
dual
graph
Γ
V
•
,
whose
vertices
correspond
bijectively
to
the
irreducible
components
of
V
•
×
k
•
k,
and
whose
edges
correspond
bijectively
to
the
[1]
points
of
(V
•
\U
V
•
)
×
k
•
k
[so
each
edge
abuts
to
the
vertices
corresponding
to
the
irreducible
components
in
which
the
point
corresponding
to
the
edge
lies].
In
dis-
cussions
of
Γ
V
•
,
we
shall
frequently
identify
the
vertices
and
edges
of
Γ
V
•
with
the
corresponding
irreducible
components
and
points
of
V
•
×
k
•
k.
If
the
natural
Galois
action
of
Gal(k/k
•
)
on
Γ
V
•
is
trivial,
then
we
shall
say
that
V
•
log
is
split.
Finally,
def
the
loop-rank
lp-rk(V
•
)
=
lp-rk(Γ
V
•
)
[cf.
§0]
is
equal
to
the
loop-rank
lp-rk(X).
(vii)
(Filtered
Projective
Systems)
Given
any
log-modification
V
◦
log
→
log
X
log
,
there
exists
a
log-modification
V
•◦
→
X
log
that
admits
X
log
-morphisms
log
log
log
log
V
•◦
→
V
•
,
V
•◦
→
V
◦
.
That
is
to
say,
the
log-modifications
over
X
log
form
a
filtered
projective
system.
(viii)
(Functoriality)
Let
Y
→
S
be
a
stable
curve,
with
smooth
generic
def
fiber
Y
η
=
Y
×
S
η;
Y
log
the
log
regular
log
scheme
determined
by
the
open
sub-
scheme
Y
η
⊆
Y
.
Then
every
finite
morphism
Y
η
→
X
η
extends
to
a
commutative
diagram
W
•
log
−→
V
•
log
⏐
⏐
⏐
⏐
Y
log
−→
X
log
—
where
W
•
log
→
Y
log
is
a
log-modification.
If
lp-rk(Y
)
=
lp-rk(X),
then
we
shall
say
that
the
morphisms
Y
η
→
X
η
,
Y
log
→
X
log
,
W
•
log
→
V
•
log
are
loop-
preserving;
if
lp-rk(Y
)
>
lp-rk(X)
[or,
equivalently,
lp-rk(Y
)
=
lp-rk(X)],
then
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
37
we
shall
say
that
the
morphisms
Y
η
→
X
η
,
Y
log
→
X
log
,
W
•
log
→
V
•
log
are
loop-
ifying.
Let
C
be
an
irreducible
component
of
W
•
.
Then
we
shall
refer
to
C
as
base-stable
[relative
to
Y
η
→
X
η
]
(respectively,
base-semi-stable
[relative
to
W
•
log
→
V
•
log
])
if
it
maps
finitely
to
a(n)
stable
(respectively,
arbitrary)
irreducible
component
of
V
•
.
If
there
exist
log-modifications
W
◦
log
→
Y
log
,
V
◦
log
→
X
log
that
fit
into
a
commutative
diagram
W
◦
log
⏐
⏐
−→
V
◦
log
⏐
⏐
W
•
log
−→
V
•
log
[where
the
left-hand
vertical
arrow
is
a
Y
log
-morphism;
the
right-hand
vertical
arrow
is
an
X
log
-morphism]
such
that
C
is
the
image
of
a
base-semi-stable
[relative
to
W
◦
log
→
V
◦
log
]
irreducible
component
of
W
◦
,
then
we
shall
say
that
C
is
potentially
base-semi-stable
[relative
to
Y
η
→
X
η
].
(ix)
(Centers
in
the
1-Interior)
Let
k
◦
be
a
finite
separable
extension
of
k;
K
◦
a
discretely
valued
field
containing
k
◦
which
induces
an
inclusion
O
k
◦
⊆
O
K
◦
∼
×
between
between
the
respective
rings
of
integers
and
a
bijection
k
◦
/O
k
×
◦
→
K
◦
/O
K
◦
the
respective
value
groups;
x
◦
∈
X(K
◦
)
a
K
◦
-valued
point.
Then
there
exists
an
unramified
log-modification
V
◦
log
→
X
log
with
base-field
k
◦
such
that
the
def
morphism
V
◦
→
S
◦
=
Spec(O
k
◦
)
is
a
semi-stable
curve,
and
x
◦
extends
to
a
[1]
point
∈
U
V
◦
(O
K
◦
).
(x)
(Maps
to
the
Jacobian)
Suppose
[for
simplicity]
that
φ
log
is
a
base-
•
[1]
field-isomorphism.
Let
x
•
∈
U
V
•
(O
k
);
C
the
[unique,
by
(vi)]
irreducible
compo-
[1]
def
nent
of
V
•
that
meets
[the
image
of
]
x
•
;
F
x
•
=
V
•
\(C
U
V
•
)
⊆
V
•
[regarded
as
a
def
closed
subset];
U
x
•
=
V
•
\F
x
•
⊆
V
•
[so
the
image
of
x
•
lies
in
U
x
•
].
Write
J
η
→
η
for
the
Jacobian
of
X
η
;
J
→
S
for
the
uniquely
determined
semi-abelian
scheme
over
S
that
extends
J
η
;
ι
η
:
X
η
→
J
η
for
the
morphism
that
sends
a
T
-
valued
point
ξ
[where
T
is
a
k-scheme]
of
X
η
,
regarded
as
a
divisor
on
X
η
×
k
T
,
to
the
point
of
J
η
determined
by
the
degree
zero
divisor
ξ
−
(x
•
|
T
).
Then
ι
η
extends
uniquely
to
a
morphism
U
x
•
→
J.
If,
moreover,
X
is
loop-ample
[cf.
§0],
then
this
morphism
U
x
•
→
J
is
unramified.
(xi)
(Lifting
Simple
Paths)
In
the
situation
of
(viii),
suppose
further
that
the
following
conditions
hold:
(a)
the
log-modifications
W
•
log
→
Y
log
,
V
•
log
→
X
log
are
base-field-isomorphic
and
split
[cf.
(vi)];
(b)
the
morphism
W
•
log
→
V
•
log
is
finite.
Let
γ
V
be
a
simple
path
[cf.
§0]
in
the
dual
graph
Γ
V
•
of
V
•
[cf.
(vi)].
Then
there
exists
a
simple
path
γ
W
in
the
dual
graph
Γ
W
•
of
W
•
that
lifts
γ
V
in
the
∼
sense
that
the
morphism
W
•
→
V
•
induces
an
isomorphism
of
graphs
γ
W
→
γ
V
.
Suppose
further
that
the
following
condition
holds:
(c)
the
morphism
W
•
log
→
V
•
log
is
loop-preserving.
38
SHINICHI
MOCHIZUKI
Then
γ
W
is
unique
in
the
sense
that
if
γ
W
is
any
simple
path
in
Γ
W
•
that
lifts
γ
V
and
is
co-terminal
[cf.
§0]
with
γ
W
,
then
γ
W
=
γ
W
.
(xii)
(Loop-preservation
and
Wild
Ramification)
In
the
situation
of
(xi),
suppose
that,
in
addition
to
the
conditions
(a),
(b),
(c)
of
(xi),
the
following
con-
ditions
hold:
(d)
there
exists
a
prime
number
p
such
that
k
is
of
characteristic
p,
and
the
morphism
Y
η
→
X
η
is
finite
étale
Galois
and
of
degree
p;
(e)
the
morphism
Y
η
→
X
η
is
wildly
ramified
over
the
terminal
vertices
[cf.
§0]
of
the
simple
path
γ
V
.
Let
w
exc
be
a
vertex
of
Γ
W
•
that
corresponds
to
an
irreducible
component
of
the
exceptional
divisor
of
W
•
log
→
Y
log
[i.e.,
a
non-stable
irreducible
component
of
W
•
]
and,
moreover,
maps
to
a
vertex
v
exc
of
Γ
V
•
lying
in
γ
V
.
Then
the
morphism
Y
η
→
X
η
is
wildly
ramified
at
w
exc
.
Proof.
First,
we
consider
assertion
(i).
We
may
assume
without
loss
of
generality
that
ψ
is
a
base-field-isomorphism.
Then
it
follows
from
the
simple
structure
of
the
[1]
monoid
N
that
any
log
étale
birational
morphism
over
U
V
1
is,
in
fact,
étale.
This
completes
the
proof
of
assertion
(i).
Next,
we
consider
assertion
(iii).
Since
the
morphism
X
log
→
S
log
is
log
smooth,
and
the
morphism
V
•
log
→
X
log
×
S
log
S
•
log
is
log
étale,
we
conclude
that
V
•
log
is
log
smooth
over
S
•
log
;
the
portion
of
assertion
(iii)
concerning
ω
V
log
/S
log
then
follows
immediately.
To
verify
the
portion
of
assertion
•
•
(iii)
concerning
unramified
log-modifications,
it
suffices
to
observe
that,
in
light
of
the
well-known
local
structure
of
the
nodes
of
the
stable
log
curve
X
log
×
S
log
S
•
log
→
def
S
•
log
,
there
exists
a
finite
extension
k
◦
of
k
•
such
that,
if
we
write
S
◦
=
Spec(O
k
◦
),
def
then
the
morphism
V
◦
log
=
V
•
log
×
S
log
S
◦
log
→
S
◦
log
admits
sections
that
intersect
•
[1]
with
every
irreducible
component
of
V
◦
U
V
◦
;
thus,
the
fact
that
V
◦
log
→
X
log
is
an
unramified
log-modification
follows
immediately
from
the
log
smoothness
of
[1]
U
V
•
,
in
light
of
the
simple
structure
of
the
monoid
N.
This
completes
the
proof
of
assertion
(iii).
Next,
we
consider
assertion
(iv).
The
fact
that
V
•
log
is
log
regular
follows
immediately
from
the
log
smoothness
of
V
•
log
over
S
•
log
[cf.
assertion
(iii)];
the
fact
[1]
[1]
that
U
V
•
is
regular
then
follows
from
the
log
regularity
of
U
V
•
,
in
light
of
the
simple
structure
of
the
monoid
N.
To
construct
a
regular
log-modification
V
◦
log
→
X
log
that
admits
a
base-field-isomorphic
X
log
-morphism
V
◦
log
→
V
•
log
,
it
suffices
to
“resolve
[1]
the
singularities”
at
the
finitely
many
points
of
V
•
\U
V
•
.
To
give
a
“resolution
of
singularities”
of
the
sort
desired,
it
suffices
to
construct,
for
each
such
v,
a
“fan”
arising
from
a
“locally
finite
nonsingular
subdivision
of
the
strongly
convex
rational
polyhedral
cone
associated
to
the
stalk
of
the
characteristic
sheaf
of
V
•
log
at
v
that
is
equivariant
with
respect
to
the
Galois
action
on
the
stalk”
[cf.,
e.g.,
the
discussion
at
the
beginning
of
[Mzk2],
§2].
Since
this
is
always
possible
[cf.,
e.g.,
the
references
quoted
in
the
discussion
of
loc.
cit.],
we
thus
obtain
a
regular
log-modification
V
◦
log
→
X
log
that
admits
a
base-field-isomorphic
X
log
-morphism
V
◦
log
→
V
•
log
;
moreover,
by
replacing
V
◦
log
with
the
result
of
blowing
up
once
more
at
[1]
various
points
of
V
◦
\U
V
◦
,
we
may
assume
that
each
irreducible
component
C
of
V
◦
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
39
is
smooth
over
k
◦
,
as
desired.
Finally,
the
construction
of
the
natural
isomorphism
[1]
∼
ω
C/k
◦
(D
C
)
→
ω
V
log
/S
log
|
C
is
immediate
over
C
U
V
◦
[cf.
our
assumption
that
the
◦
◦
log-modification
V
◦
log
→
X
log
is
unramified
at
C!];
one
may
then
extend
this
natural
isomorphism
to
C
by
means
of
an
easy
local
calculation
at
the
points
of
D
C
.
This
completes
the
proof
of
assertion
(iv).
By
assertions
(iii)
and
(iv),
the
underlying
schemes
of
the
domain
and
codomain
of
the
morphism
V
2
log
→
V
1
log
×
S
log
S
2
log
of
1
assertion
(ii)
are
normal.
Thus,
assertion
(ii)
follows
immediately
from
Zariski’s
main
theorem.
Next,
we
consider
assertion
(v).
We
may
assume
without
loss
of
generality
are
base-field-isomorphic
[so
ψ
log
is
log
étale].
Also,
by
blowing
up
that
the
φ
log
i
[1]
once
more
at
various
points
of
V
2
\U
V
2
[cf.
the
proof
of
assertion
(iv)],
one
verifies
immediately
that
we
may
assume
without
loss
of
generality
that
C
is
smooth
over
k
2
.
Let
us
write
C
log
for
the
log
scheme
obtained
by
equipping
C
with
the
log
structure
determined
by
the
points
of
C
that
meet
the
other
irreducible
components
of
D
V
2
.
Thus,
after
possibly
replacing
k
2
by
a
suitable
finite
unramified
extension
of
k
2
,
we
may
assume
that
the
interior
U
C
⊆
C
of
C
log
is
the
open
subscheme
of
a
smooth
proper
curve
of
genus
g
over
k(c)
obtained
by
removing
a
divisor
D
C
⊆
C
of
degree
r
>
0
over
k(c).
On
the
other
hand,
it
follows
immediately
from
the
definition
of
a
log-modification
[cf.
Definition
2.4,
(i)]
and
the
well-known
general
theory
of
toric
varieties
[cf.
the
discussion
of
“fans”
in
the
proof
of
assertion
(iv)]
that
the
pair
(C,
D
C
)
determines
a
toric
variety
of
dimension
one,
and
hence
that
U
C
is
isomorphic
to
a
copy
of
G
m
over
k(c),
i.e.,
that
g
=
0,
r
=
2,
as
desired.
Now
the
remaining
portions
of
assertion
(v)
follow
immediately.
This
completes
the
proof
of
assertion
(v).
Next,
we
consider
assertion
(vi).
First,
we
observe
that
if
x
belongs
to
the
1-interior,
then
it
follows
immediately
from
the
simple
structure
of
the
monoid
N
[and
the
definition
of
the
1-interior]
that
V
x
is
irreducible.
Thus,
it
suffices
to
consider
the
case
where
x
does
not
belong
to
the
1-interior.
Let
V
◦
log
→
X
log
,
V
◦
log
→
V
•
log
be
as
in
assertion
(iv).
We
may
assume
without
loss
of
generality
that
the
log-modifications
V
•
log
→
X
log
,
V
◦
log
→
X
log
are
base-field-isomorphisms.
Also,
by
replacing
k
be
a
finite
unramified
extension
of
k,
we
may
assume
that
every
irreducible
component
of
V
•
is
geometrically
irreducible
over
k,
and
that
every
point
[1]
of
V
•
\U
V
•
is
defined
over
k.
Then,
since
V
◦
log
→
V
•
log
is
birational,
and
V
•
log
is
log
regular,
hence,
in
particular,
normal
[cf.
assertion
(iv)],
it
follows
from
Zariski’s
[1]
main
theorem
that
the
points
of
V
•
\U
V
•
correspond
precisely
[via
V
◦
log
→
V
•
log
]
to
[1]
the
connected
components
of
the
inverse
image
of
V
•
\U
V
•
via
V
◦
log
→
V
•
log
.
Thus,
the
remainder
of
assertion
(vi)
follows
immediately
from
assertion
(v),
applied
to
V
◦
log
→
V
•
log
,
V
◦
log
→
X
log
.
This
completes
the
proof
of
assertion
(vi).
Next,
we
consider
assertion
(vii).
We
may
assume
without
loss
of
generality
that
the
log-modifications
V
•
log
→
X
log
,
V
◦
log
→
X
log
are
base-field-isomorphic.
log
def
Then
to
verify
assertion
(vii),
it
suffices
to
observe
that
one
may
take
V
•◦
=
log
log
V
•
×
X
log
V
◦
.
This
completes
the
proof
of
assertion
(vii).
In
a
similar
vein,
assertion
(viii)
follows
by
observing
that
the
fact
that
Y
η
→
X
η
extends
to
a
morphism
Y
log
→
X
log
follows,
for
instance,
from
[Mzk2],
Theorem
A,
(1);
thus,
def
one
may
take
W
•
log
=
V
•
log
×
X
log
Y
log
.
This
completes
the
proof
of
assertion
(viii).
40
SHINICHI
MOCHIZUKI
Next,
we
consider
assertion
(ix).
We
may
assume
without
loss
of
generality
that
k
=
k
◦
.
Let
V
◦
log
→
X
log
be
the
base-field-isomorphic
unramified
log-modification
determined
by
the
regular
semi-stable
model
of
X
over
S
[cf.
Remark
2.4.1].
Since
V
◦
is
proper
over
S,
it
follows
that
x
◦
extends
to
a
point
∈
V
◦
(O
K
◦
);
if
this
point
[1]
fails
to
lie
in
U
V
◦
,
then
it
follows
that
it
meets
one
of
the
nodes
of
V
◦
.
On
the
other
hand,
since
[after
possibly
replacing
k
◦
by
a
finite
unramified
extension
of
the
discretely
valued
field
k
◦
]
the
completion
of
the
regular
scheme
V
◦
at
such
a
node
is
necessarily
isomorphic
over
O
k
◦
to
a
complete
local
ring
of
the
form
O
k
◦
[[s,
t]]/(s
·
t
−
π
◦
)
[where
s,
t
are
indeterminates;
π
◦
is
a
uniformizer
of
O
k
◦
],
this
contradicts
our
∼
×
assumption
that
O
k
◦
⊆
O
K
◦
induces
a
bijection
k
◦
/O
k
×
◦
→
K
◦
/O
K
[i.e.,
by
con-
◦
sidering
the
images
via
pull-back
by
x
◦
of
s,
t
in
these
value
groups,
in
light
of
the
relation
“s
·
t
−
π
◦
”].
This
completes
the
proof
of
assertion
(ix).
Next,
we
consider
assertion
(x).
Write
N
→
S
for
the
Néron
model
of
J
η
over
S.
Thus,
J
may
be
regarded
as
an
open
subscheme
of
N
.
Note
that
the
existence
of
the
rational
point
x
•
implies
that
U
x
•
is
smooth
over
S
[cf.
the
proof
of
assertion
(iii)].
Thus,
it
follows
from
the
universal
property
of
the
Néron
model
[which
is
typically
used
to
define
the
Néron
model]
that
ι
η
extends
to
a
morphism
U
x
•
→
N
.
Since
C
U
x
•
is
connected
[cf.
the
definition
of
U
x
•
],
the
fact
that
the
image
of
this
morphism
lies
in
J
⊆
N
follows
immediately
from
the
fact
that
[by
definition]
it
maps
x
•
to
the
identity
element
of
J(O
k
).
Thus,
we
obtain
a
morphism
U
x
•
→
J.
To
verify
that
this
morphism
is
unramified,
it
suffices
[by
considering
appropriate
translation
automorphisms
of
J]
to
show
that
it
induces
a
surjection
on
Zariski
cotangent
spaces
at
x
•
;
but
the
induced
map
on
Zariski
cotangent
spaces
at
x
•
is
easily
computed
[by
considering
the
long
exact
sequence
on
cohomology
associated
to
the
short
exact
sequence
0
→
O
V
•
→
O
V
•
(x
•
)
→
O
V
•
(x
•
)|
x
•
→
0
on
V
•
,
then
taking
duals]
to
be
the
map
∼
H
0
(X
log
,
ω
X
log
/S
log
)
→
H
0
(V
log
,
ω
V
log
/S
log
)
→
ω
V
log
/S
log
|
x
•
•
•
∼
[where
we
recall
the
natural
isomorphism
ω
X
log
/S
log
|
V
log
→
ω
V
log
/S
log
,
arising
from
•
•
log
→
X
log
is
log
étale
—
cf.
assertion
(iii)]
given
by
evaluating
the
fact
that
φ
log
•
:
V
•
at
x
•
,
hence
is
surjective
so
long
as
X
is
loop-ample
[cf.
§0].
This
completes
the
proof
of
assertion
(x).
Next,
we
consider
assertion
(xi).
First,
let
us
observe
that
it
follows
immedi-
ately
from
the
surjectivity
of
W
•
→
V
•
that
every
vertex
of
Γ
V
•
may
be
lifted
to
a
vertex
of
Γ
W
•
.
Next,
I
claim
that
every
edge
of
Γ
V
•
may
be
lifted
to
an
edge
of
Γ
W
•
.
Indeed,
let
y
∈
W
•
(k),
x
∈
V
•
(k)
be
such
that
y
→
x;
write
W
y
,
V
x
for
the
respective
spectra
of
the
local
rings
obtained
by
completing
W
•
,
V
•
at
y,
x.
Then
the
morphism
W
log
→
V
log
induces
a
finite,
dominant,
hence
surjective,
morphism
W
y
→
V
x
.
In
particular,
this
morphism
W
y
→
V
x
induces
a
surjection
from
the
set
I
y
of
irreducible
components
of
W
•
that
pass
through
y
to
the
set
I
x
of
irreducible
components
of
V
•
that
pass
through
x.
Thus,
if
x
corresponds
to
an
edge
of
the
dual
graph
Γ
V
•
,
then
this
set
I
x
is
of
cardinality
2
[cf.
assertion
(vi)];
since
I
y
is
of
cardinality
≤
2
[cf.
assertion
(vi)],
the
existence
of
the
surjection
I
y
I
x
thus
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
41
∼
implies
that
this
surjection
is,
in
fact,
a
bijection
I
y
→
I
x
,
hence
that
y
corresponds
to
an
edge
of
Γ
W
•
[cf.
assertion
(vi)].
This
completes
the
proof
of
the
claim.
Thus,
by
starting
at
one
of
the
terminal
vertices
of
γ
V
,
and
proceeding
along
γ
V
from
vertex
to
edge
to
vertex,
etc.,
one
concludes
immediately
the
existence
of
a
simple
path
γ
W
lifting
γ
V
.
To
verify
uniqueness
when
condition
(c)
holds,
write
v
1
,
v
2
for
the
terminal
vertices
of
γ
V
;
e
for
the
edge
of
γ
V
that
is
nearest
to
v
1
among
those
;
v
1
(respectively,
v
2
)
for
the
vertex
edges
of
γ
V
that
lift
to
different
edges
in
γ
W
,
γ
W
to
which
e
abuts
that
lies
in
the
same
connected
component
of
the
complement
of
e
in
γ
V
as
v
1
(respectively,
v
2
);
v
1
+
for
the
vertex
of
γ
V
that
is
nearest
to
v
1
among
those
vertices
of
γ
V
lying
between
v
2
and
v
2
which
lift
to
the
same
vertex
in
γ
W
,
.
Then
by
traveling
along
γ
W
from
the
vertex
w
1
of
γ
W
lifting
v
1
to
the
vertex
γ
W
+
w
1
of
γ
W
lifting
v
1
+
,
then
traveling
back
along
γ
W
from
w
1
+
[which,
by
definition,
]
to
w
1
[which,
by
definition,
also
belongs
to
γ
W
],
one
obtains
also
belongs
to
γ
W
a
“nontrivial
loop”
in
Γ
W
•
[i.e.,
a
nonzero
element
of
H
1
(Γ
W
•
,
Q)]
that
maps
to
a
“trivial
loop”
in
Γ
V
•
[i.e.,
the
zero
element
of
H
1
(Γ
V
•
,
Q)].
But
this
contradicts
the
assumption
that
the
morphism
W
•
log
→
V
•
log
is
loop-preserving
[cf.
condition
(c)].
This
completes
the
proof
of
assertion
(xi).
Finally,
we
consider
assertion
(xii).
First,
we
observe
that
the
hypotheses
of
assertion
(xii)
are
stable
with
respect
to
base-change
in
S.
In
particular,
we
may
always
replace
S
=
Spec(O
k
)
by
the
normalization
of
S
in
some
finite
separable
extension
of
k.
Next,
by
assertion
(iv),
we
may
assume
that
there
exists
a
base-
field-isomorphic,
regular,
split
log-modification
V
◦
log
→
X
log
,
together
with
an
X
log
-
def
morphism
V
◦
log
→
V
•
log
.
Moreover,
if
we
take
W
◦
log
=
V
◦
log
×
V
log
W
•
log
,
then
•
the
composite
morphism
W
◦
log
→
W
•
log
→
Y
log
forms
a
base-field-isomorphic
log-
modification
such
that
the
projection
W
◦
log
→
V
◦
log
is
finite
[by
condition
(b);
cf.
also
the
finiteness
mentioned
in
the
discussion
entitled
“Log
Schemes”
given
in
§0].
In
particular,
by
replacing
W
•
log
→
V
•
log
by
W
◦
log
→
V
◦
log
[cf.
also
assertion
(v),
concerning
the
effect
on
the
simple
path
γ
V
;
assertion
(vi),
concerning
the
effect
on
the
loop-rank],
we
may
assume
that
the
log-modification
V
•
log
→
X
log
is
regular.
Here,
let
us
note
that
since
W
•
log
→
V
•
log
is
finite,
and
W
•
log
is
log
regular
[cf.
assertion
(iv)],
which
implies,
in
particular,
that
W
•
is
normal
[so
W
•
is
the
def
log
normalization
of
V
•
in
Y
η
],
it
follows
that
G
=
Gal(Y
η
/X
η
)
(
∼
=
Z/pZ)
acts
on
W
•
.
Also,
by
assertion
(iv),
we
may
assume
that
there
exists
a
base-field-isomorphic,
log
log
regular,
split
log-modification
W
→
Y
log
,
together
with
a
Y
log
-morphism
W
→
W
•
log
;
moreover,
it
follows
immediately
from
the
proof
of
assertion
(iv)
that
we
log
log
may
choose
W
so
that
the
action
of
G
extends
to
W
.
Finally,
we
observe
that
it
follows
from
assertion
(v)
that
every
irreducible
component
of
the
exceptional
log
divisors
of
W
,
V
•
[relative
to
the
morphisms
W
→
Y
log
,
V
•
log
→
X
log
]
is
isomorphic
to
P
1
k
.
Let
E
W
be
the
irreducible
component
of
W
•
corresponding
to
w
exc
.
Thus,
there
exists
a
unique
irreducible
component
F
W
of
W
that
maps
finitely
to
E
W
;
moreover,
E
W
maps
finitely
to
an
irreducible
component
E
V
of
V
•
[corresponding
to
v
exc
]
that
lies
in
the
exceptional
divisor
of
V
•
log
→
X
log
.
Thus,
we
have
finite
morphisms
F
W
→
E
W
→
E
V
42
SHINICHI
MOCHIZUKI
—
where
F
W
,
E
V
are
isomorphic
to
P
1
k
;
the
first
morphism
F
W
→
E
W
is
a
mor-
phism
between
irreducible
schemes
that
induces
an
isomorphism
between
the
re-
spective
function
fields.
Now
to
complete
the
proof
of
assertion
(xii),
it
suffices
to
assume
that
the
composite
morphism
F
W
→
E
V
is
generically
étale
and
derive
a
contradiction.
Let
us
refer
to
the
two
k-valued
points
of
F
W
(respec-
[1]
[1]
tively,
E
V
)
[cf.
assertion
(v)]
that
lie
outside
U
W
(respectively,
U
V
•
)
as
the
critical
points
of
F
W
(respectively,
E
V
).
Then
since
the
divisor
on
W
•
log
(respectively,
V
•
log
)
at
which
the
morphism
W
•
log
→
V
•
log
is
[necessarily
wildly]
ramified
does
not,
by
our
assumption,
contain
E
W
(respectively,
E
V
),
it
follows
that
the
divisor
on
F
W
(re-
spectively,
E
V
)
at
which
F
W
→
E
V
is
ramified
is
supported
in
the
divisor
defined
by
the
two
critical
points
of
F
W
(respectively,
E
V
).
Next,
let
us
write
v
1
,
v
2
for
the
terminal
vertices
of
γ
V
.
Note
that
by
conditions
(d),
(e),
it
follows
that
v
1
,
v
2
lift,
respectively,
to
unique
vertices
w
1
,
w
2
of
Γ
W
•
.
In
particular,
it
follows
that
any
two
simple
paths
in
Γ
W
•
lifting
γ
V
are
co-terminal.
Now
I
claim
that
the
morphism
F
W
→
E
V
is
of
degree
p.
Indeed,
if
this
morphism
of
w
exc
such
that
is
of
degree
1,
then
it
follows
that
there
exists
a
G-conjugate
w
exc
w
exc
=
w
exc
.
Thus,
considering
the
G-conjugates
of
any
simple
path
in
Γ
W
•
lifting
γ
V
,
it
follows
that
we
obtain
two
distinct
[necessarily
co-terminal!]
simple
paths
in
Γ
W
•
lifting
γ
V
.
But
this
contradicts
the
uniqueness
portion
of
assertion
(xi).
This
completes
the
proof
of
the
claim.
Note
that
this
claim
implies
that
we
have
G-equivariant
morphisms
F
W
→
E
W
→
E
V
,
where
G
acts
trivially
on
E
V
.
Next,
I
claim
that
G
fixes
each
of
the
critical
points
of
F
W
.
Indeed,
it
follows
immediately
from
the
definitions
that
G
preserves
the
divisor
of
critical
points
of
F
W
.
Thus,
if
G
fails
to
fix
each
of
the
critical
points
of
F
W
,
then
it
follows
that
G
permutes
the
two
critical
points
of
F
W
,
hence
that
p
=
2.
But
since
F
W
→
E
V
is
of
degree
p
and
unramified
outside
the
divisor
of
critical
points
of
F
W
,
this
implies
that
P
1
k
∼
=
F
W
→
E
V
∼
=
P
1
k
is
finite
étale,
hence
[since,
as
is
well-known,
the
étale
fundamental
group
of
P
1
k
is
trivial!]
that
F
W
→
E
V
is
an
isomorphism
—
in
contradiction
to
the
fact
that
F
W
→
E
V
is
of
degree
p
>
1.
This
completes
the
proof
of
the
claim.
Note
that
this
claim
implies
that
the
morphism
F
W
→
E
V
is
ramified
at
the
critical
points
of
F
W
,
and
that
the
set
of
two
critical
points
of
F
W
maps
bijectively
to
the
set
of
two
critical
points
of
E
V
.
In
particular,
it
follows
that
F
W
→
E
V
determines
a
finite
étale
covering
(G
m
)
k
→
(G
m
)
k
of
degree
p.
On
the
other
hand,
any
morphism
(G
m
)
k
→
(G
m
)
k
is
determined
by
a
unit
on
(G
m
)
k
,
i.e.,
by
a
k
×
-multiple
of
U
n
,
where
U
is
the
standard
coordinate
on
(G
m
)
k
,
and
n
is
the
degree
of
the
morphism.
Since
the
morphism
determined
by
a
k
×
-multiple
of
U
p
clearly
fails
to
be
generically
étale,
we
thus
obtain
a
contradiction.
This
completes
the
proof
of
assertion
(xii).
In
the
following,
we
shall
write
“π
1
(−)”
for
the
“log
fundamental
group”
of
the
log
scheme
in
parentheses,
relative
to
an
appropriate
choice
of
basepoint
[cf.
[Ill]
for
a
survey
of
the
theory
of
log
fundamental
groups].
Also,
from
now
on,
we
shall
assume,
until
further
notice,
that:
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
43
The
discrete
valuation
ring
O
k
is
of
mixed
characteristic,
with
residue
field
k
perfect
of
characteristic
p
and
of
countable
cardinality.
Recall
that
by
“Krasner’s
lemma”
[cf.
[Kobl],
pp.
69-70,
as
well
as
the
proof
given
above
of
Lemma
2.1],
given
a
splitting
field
k
over
k
of
a
monic
polynomial
f
(T
)
[where
T
is
an
indeterminate]
of
degree
n
with
coefficients
in
k,
every
monic
polynomial
h(T
)
of
degree
n
with
coefficients
in
k
that
are
sufficiently
close
[in
the
topology
of
k]
to
the
coefficients
of
f
(T
)
also
splits
in
k
.
Thus,
it
follows
from
our
assumption
that
k
is
of
countable
cardinality
that
k
admits
a
countable
collection
F
of
subfields
which
are
finite
and
Galois
over
k
such
that
every
finite
Galois
extension
of
k
contained
in
k
is
contained
in
a
subfield
that
belongs
to
the
collection
F.
In
some
sense,
the
main
technical
result
of
the
present
§2
is
the
following
lemma.
Lemma
2.6.
(Prime-power
Cyclic
Coverings
and
Log-modifications)
Suppose
that
X
is
loop-ample
[cf.
§0].
Then:
(i)
(Existence
of
Wild
Ramification)
Let
X
η
+
→
X
η
be
a
finite
étale
Galois
covering
of
hyperbolic
curves
over
η
with
stable
reduction
over
S
such
that
Gal(X
η
+
/X
η
)
is
isomorphic
to
a
product
of
2g
X
copies
of
Z/pZ
[so
such
a
covering
always
exists
after
possibly
replacing
k
by
a
finite
extension
of
k];
V
log
→
X
log
a
split,
base-field-isomorphic
log-modification.
Then
X
η
+
→
X
η
is
wildly
ramified
over
every
irreducible
component
C
of
V
.
(ii)
(Loopification
vs.
Component
Crushing)
After
possibly
replacing
k
by
a
finite
extension
of
k,
there
exist
data
as
follows:
a
stable
curve
Y
→
S
with
def
smooth
generic
fiber
Y
η
=
Y
×
S
η
and
associated
log
scheme
Y
log
;
a
cyclic
finite
étale
covering
Y
η
→
X
η
of
degree
a
positive
power
of
p
—
which
determines
a
morphism
Y
log
→
X
log
—
such
that
at
least
one
of
the
following
two
conditions
is
satisfied:
(a)
Y
η
→
X
η
is
loopifying
and
wildly
ramified
at
some
[necessarily]
stable
irreducible
component
C
of
Y
which
is
potentially
base-semi-
stable
relative
to
Y
η
→
X
η
;
(b)
there
exists
a
[necessarily]
stable
irreducible
component
C
of
Y
which
is
not
potentially
base-semi-stable
relative
to
Y
η
→
X
η
.
(iii)
(Components
Crushed
to
the
1-Interior)
In
the
situation
of
(ii),
there
exists
a
commutative
diagram
W
log
⏐
⏐
−→
Q
log
⏐
⏐
Y
log
−→
X
log
—
where
the
vertical
morphisms
are
split,
base-field-isomorphic
log-modifications;
the
horizontal
morphism
in
the
bottom
line
is
the
morphism
already
referred
to;
the
44
SHINICHI
MOCHIZUKI
natural
action
of
Gal(Y
η
/X
η
)
on
Y
log
extends
to
W
log
—
such
that
the
following
property
is
satisfied:
If
condition
(a)
(respectively,
(b))
of
(ii)
is
satisfied,
then
the
unique
irreducible
component
C
W
of
W
that
maps
finitely
to
the
irreducible
compo-
nent
C
of
condition
(a)
(respectively,
(b))
maps
finitely
to
Q
(respectively,
maps
[1]
to
a
closed
point
of
U
Q
).
(iv)
(Group-theoretic
Characterization
of
Crushing)
In
the
situation
of
(ii),
let
C
be
a
[necessarily]
stable
irreducible
component
of
Y
;
l
a
prime
=
p.
Write
Gal(k/k)
G
k
log
for
the
maximal
tamely
ramified
quotient;
Δ
Y
log
(respectively,
Δ
X
log
)
for
the
maximal
pro-l
quotient
of
the
kernel
of
the
natural
[outer]
surjection
π
1
(Y
log
)
G
k
log
(respectively,
π
1
(X
log
)
G
k
log
);
Δ
C
⊆
Δ
Y
log
for
the
decomposition
group
of
C
[well-defined
up
to
conjuga-
tion
by
an
element
of
π
1
(Y
log
)].
[Thus,
Δ
Y
log
(respectively,
Δ
X
log
)
may
be
identi-
fied
with
the
maximal
pro-l
quotient
of
Ker(π
1
(Y
η
)
Gal(k/k))
(respectively,
Ker(π
1
(X
η
)
Gal(k/k)))
—
cf.,
e.g.,
[MT],
Proposition
2.2,
(v).]
Then
the
fol-
lowing
two
conditions
are
equivalent:
(a)
the
image
of
Δ
C
in
Δ
X
log
is
trivial;
(b)
there
exists
a
commutative
diagram
W
log
⏐
⏐
−→
Q
log
⏐
⏐
Y
log
−→
X
log
—
where
the
vertical
morphisms
are
split,
base-field-isomorphic
log-modi-
fications;
the
horizontal
morphism
in
the
bottom
line
is
the
morphism
already
referred
to
—
such
that
the
unique
irreducible
component
C
W
of
[1]
W
that
maps
finitely
to
C
maps
to
a
closed
point
of
U
Q
.
(v)
(Group-theoretic
Characterization
of
Wild
Ramification)
In
the
situation
of
(iv),
the
morphism
Y
η
→
X
η
is
wildly
ramified
at
C
if
and
only
if
Gal(Y
η
/X
η
)
stabilizes
[the
conjugacy
class
of
]
and
induces
the
identity
[outer
automorphism]
on
the
[normally
terminal
—
cf.
[Mzk13],
Proposition
1.2,
(ii);
[Mzk17],
Lemma
2.12]
subgroup
Δ
C
of
Δ
Y
log
.
Proof.
Let
us
write
def
T
X
=
π
1
(X
η
×
k
k)
ab
⊗
Z
p
for
the
maximal
pro-p
abelian
quotient
of
the
geometric
fundamental
group
of
X
η
.
Thus,
T
X
is
a
free
Z
p
-module
of
rank
2g
X
.
Next,
we
consider
assertion
(i).
Upon
base-change
to
k,
the
covering
X
η
+
→
X
η
corresponds
to
the
open
subgroup
p
·
T
X
⊆
T
X
.
Let
us
write
J
→
S
for
the
uniquely
determined
semi-abelian
scheme
that
extends
the
Jacobian
J
η
→
η
of
X
η
.
After
possibly
replacing
k
by
a
finite
extension
of
k,
there
exists
a
rational
[1]
point
x
∈
U
V
(O
k
)
that
meets
C.
Thus,
for
some
Zariski
open
neighborhood
U
x
of
the
image
of
x
in
V
,
we
obtain
a
morphism
ι
:
U
x
→
J,
as
in
Proposition
2.5,
(x).
Moreover,
since
we
have
assumed
that
X
is
loop-ample,
it
follows
that
this
morphism
ι
is
unramified.
Now
if
the
morphism
X
η
+
→
X
η
is
tamely
ramified
over
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
45
C,
then
it
follows
from
our
assumptions
on
the
covering
X
η
+
→
X
η
,
together
with
the
interpretation
of
J
in
terms
of
Néron
models
[cf.
the
proof
of
Proposition
2.5,
(x)],
that
[after
possibly
replacing
k
by
a
finite
extension
of
k]
there
exists
some
finite
separable
extension
L
of
the
function
field
k(C)
of
C
such
that
there
exists
a
commutative
diagram
ι
L
−→
J
Spec(L)
⏐
⏐
⏐
⏐
[p]
Spec(k(C))
ι
C
−→
J
—
where
is
the
[étale]
morphism
determined
by
the
given
inclusion
k(C)
→
L;
[p]
is
the
morphism
given
by
multiplication
by
p
on
the
group
scheme
J;
ι
C
is
the
restriction
of
ι
to
Spec(k(C)).
On
the
other
hand,
since
the
restriction
of
[p]
to
the
special
fiber
J
of
J
factors
through
the
Frobenius
morphism
on
J,
it
follows
that
[p]
◦
ι
L
fails
to
be
unramified.
Thus,
since
ι
C
◦
is
unramified,
we
obtain
a
contradiction
to
the
commutativity
of
the
diagram.
This
completes
the
proof
of
assertion
(i).
Next,
we
consider
assertion
(ii).
If
V
log
→
X
log
is
a
split,
base-field-isomorphic
log-modification,
and
C
is
an
irreducible
component
of
V
,
then
let
us
write
D
C
⊆
T
X
for
the
decomposition
group
associated
to
C
and
I
C
⊆
D
C
for
the
wild
inertia
group
associated
to
C.
Note
that
since
T
X
is
abelian,
and
V
log
is
split,
it
follows
that
the
subgroups
D
C
,
I
C
are
well-defined
[and
completely
determined
by
C].
Moreover,
by
assertion
(i),
it
follows
that
I
C
has
nontrivial
image
in
T
X
⊗
Z/pZ.
Next,
I
claim
that
if
V
◦
log
→
X
log
is
a
split,
base-field-isomorphic
log-modification,
then
V
◦
log
→
X
log
is
completely
determined,
as
a
log
scheme
over
X
log
,
up
to
count-
ably
many
possibilities,
by
X
log
.
Indeed,
the
morphism
V
◦
log
→
X
log
is
an
iso-
morphism
over
the
1-interior
of
X
log
[cf.
Proposition
2.5,
(i),
(ii)].
Moreover,
at
each
of
the
finitely
many
points
x
of
X
log
lying
in
the
complement
of
the
1-interior,
V
◦
log
→
X
log
is
determined
by
countably
many
choices
of
certain
combinatorial
data
involving
the
groupification
of
the
stalk
of
the
characteristic
sheaf
of
X
log
at
x
[cf.
the
proof
of
Proposition
2.5,
(iv)].
This
completes
the
proof
of
the
claim.
In
partic-
ular,
since
k
is
assumed
to
be
of
countable
cardinality
[cf.
the
discussion
preceding
the
present
Lemma
2.6],
it
follows
that:
There
exists
a
countable
cofinal
collection
M
of
split
log-modifications
of
X
log
.
In
particular,
it
follows
that
if
we
write
C
for
the
set
of
all
irreducible
components
of
the
special
fibers
of
log-modifications
belonging
to
M,
then
the
collection
of
[non-
trivial]
subgroups
of
T
X
of
the
form
“I
C
”,
where
C
∈
C,
is
of
countable
cardinality.
Thus,
we
may,
for
instance,
enumerate
the
elements
of
C
via
the
natural
numbers
46
SHINICHI
MOCHIZUKI
so
as
to
obtain
a
sequence
C
1
,
C
2
,
.
.
.
[i.e.,
which
includes
all
elements
of
C].
Since
Z
p
,
on
the
other
hand,
is
of
uncountable
cardinality,
we
thus
conclude
that
there
exists
a
surjection
Λ
:
T
X
Λ
X
—
where
Λ
X
∼
=
Z
p
—
such
that
the
following
properties
are
satisfied:
def
(1)
For
every
subgroup
I
C
,
where
C
∈
C,
we
have
Λ
C
=
Λ(I
C
)
=
{0}.
(2)
There
exists
a
stable
component
C
0
of
X
such
that
Λ
C
0
=
Λ
X
.
[For
instance,
by
applying
the
fact
that
each
I
C
n
has
nontrivial
image
in
T
X
⊗
Z/pZ,
one
may
construct
Λ
by
constructing
inductively
on
n
[a
natural
number]
an
increasing
sequence
of
natural
numbers
m
1
<
m
2
<
.
.
.
such
that
I
C
n
maps
to
a
nonzero
subgroup
of
Λ
X
⊗
Z/p
m
n
Z.]
Let
us
refer
to
a
connected
finite
étale
Galois
covering
X
η
→
X
η
[which
may
only
be
defined
after
possibly
replacing
k
by
a
finite
extension
of
k]
of
hyperbolic
curves
over
η
with
stable
reduction
over
S
as
a
Λ-covering
if
the
covering
X
η
×
k
k
→
X
η
×
k
k
arises
from
an
open
subgroup
of
Λ
X
;
thus,
Gal(X
η
/X
η
)
may
be
thought
of
as
a
finite
quotient
of
Λ
X
by
an
open
subgroup
Λ
X
⊆
Λ
X
.
Note
that
since
every
Λ
C
,
where
C
∈
C,
is
isomorphic
to
Z
p
,
it
follows
that
the
following
property
also
holds:
(3)
For
any
pair
X
η
→
X
η
→
X
η
of
Λ-coverings
of
X
η
such
that
Λ
C
∼
has
nontrivial
image
in
Gal(X
η
/X
η
)
=
Λ
X
/Λ
X
,
it
follows
that
Λ
C
Λ
X
surjects
onto
Λ
X
/Λ
X
—
i.e.,
that
the
covering
X
η
→
X
η
is
totally
wildly
ramified
over
any
valuation
of
the
function
field
of
X
η
whose
center
on
X
is
equal
to
the
generic
point
of
C.
Now
to
complete
the
proof
of
assertion
(ii),
it
suffices
to
derive
a
contradiction
upon
making
the
following
two
further
assumptions:
(4)
Every
Λ-covering
is
loop-preserving.
(5)
For
every
Λ-covering
X
η
→
X
η
[which
extends
to
a
morphism
(X
)
log
→
X
log
of
log
stable
curves],
there
exists
[after
possibly
replacing
k
by
a
finite
extension
of
k]
a
split,
base-field-isomorphic
log-modification
V
log
→
X
log
such
that
the
morphism
X
η
→
X
η
∼
=
V
η
⊆
V
extends
to
a
quasi-finite
morphism
from
some
Zariski
neighborhood
in
X
of
the
generic
points
of
X
to
V
.
[Indeed,
if
assumption
(4)
is
false,
then
it
follows
immediately
that
condition
(a)
of
assertion
(ii)
holds
[cf.
property
(2)];
if
assumption
(5)
is
false,
then
it
follows
imme-
diately
that
condition
(b)
of
assertion
(ii)
holds.]
Note,
moreover,
that
assumption
(5)
implies
the
following
property:
(6)
For
every
Λ-covering
X
η
→
X
η
[which
extends
to
a
morphism
(X
)
log
→
X
log
of
log
stable
curves],
there
exist
[after
possibly
replacing
k
by
a
finite
extension
of
k],
split,
base-field-isomorphic
log-modifications
(V
)
log
→
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
47
(X
)
log
,
V
log
→
X
log
,
together
with
a
finite
morphism
(V
)
log
→
V
log
lying
over
(X
)
log
→
X
log
.
def
[Indeed,
if,
in
the
notation
of
property
(5),
one
takes
(V
)
log
=
V
log
×
X
log
(X
)
log
,
then
the
natural
projection
morphism
(V
)
log
→
(X
)
log
is
a
split,
base-field-
isomorphic
log-modification.
Moreover,
every
irreducible
component
of
V
maps
finitely
either
to
some
irreducible
component
of
V
or
to
some
irreducible
compo-
nent
of
X
.
Thus,
by
property
(5),
we
conclude
that
every
irreducible
component
of
V
maps
finitely
to
some
irreducible
component
of
V
,
hence
[by
Zariski’s
main
theorem]
that
the
natural
projection
morphism
(V
)
log
→
V
log
is
finite,
as
desired.]
Next,
let
us
observe
that
properties
(4),
(6)
imply
the
following
properties:
(7)
There
exists
a
Λ-covering
X
η
→
X
η
[after
possibly
replacing
k
by
a
finite
extension
of
k]
such
that
every
Λ
C
,
where
C
∈
C,
has
nontrivial
image
in
Λ
X
/Λ
X
.
(8)
There
exist
Λ-coverings
X
η
→
X
η
→
X
η
[after
possibly
replacing
k
by
a
finite
extension
of
k]
such
that
Λ
X
/Λ
X
∼
=
Z/pZ,
and,
moreover,
for
every
Λ
C
,
where
C
∈
C,
the
intersection
Λ
C
Λ
X
surjects
onto
Λ
X
/Λ
X
.
Indeed,
by
property
(3),
it
follows
that
property
(8)
follows
immediately
from
prop-
erty
(7).
To
verify
property
(7),
we
reason
as
follows:
First,
let
X
η
†
→
X
η
∗
→
X
η
be
Λ-coverings
[which
exist
after
possibly
replacing
k
by
a
finite
extension
of
k]
such
that
Λ
X
∗
/Λ
X
†
is
of
order
p,
and
Λ
C
has
nontrivial
image
in
Λ
X
/Λ
X
∗
[which
implies
that
Λ
C
Λ
X
∗
surjects
onto
Λ
X
∗
/Λ
X
†
—
cf.
property
(3)]
for
every
stable
irreducible
component
C
of
X;
write
X
†,log
→
X
∗,log
→
X
log
for
the
resulting
morphisms
of
log
stable
curves.
[Note
that
such
Λ-coverings
exist,
precisely
because
there
are
only
finitely
many
such
stable
C.]
Let
V
†,log
→
X
†,log
,
V
∗,log
→
X
∗,log
,
V
log
→
X
log
be
split,
base-field-isomorphic
log-modifications
such
that
there
exist
finite,
loop-preserving
morphisms
V
†,log
→
V
∗,log
→
V
log
lying
over
X
†,log
→
X
∗,log
→
X
log
[cf.
properties
(4),
(6)].
[Thus,
V
†,log
,
V
∗,log
are
completely
determined
by
V
log
—
i.e.,
by
taking
the
normalization
of
V
in
X
η
†
,
X
η
∗
.]
Next,
let
us
observe
that
every
node
ν
of
X
determines
a
simple
path
γ
V
ν
in
the
dual
graph
Γ
V
[i.e.,
by
taking
the
inverse
image
of
ν
in
V
—
cf.
Proposition
2.5,
(v)],
whose
terminal
vertices
are
stable
irreducible
components
of
V
[but
whose
non-
terminal
vertices
are
non-stable
irreducible
components
of
V
].
Thus,
by
Proposition
2.5,
(xi)
[which
is
applicable,
in
light
of
properties
(4),
(6)],
it
follows
that
γ
V
ν
lifts
[uniquely
—
i.e.,
once
one
fixes
liftings
of
the
terminal
vertices]
to
simple
paths
γ
V
ν
∗
in
Γ
V
∗
,
γ
V
ν
†
in
Γ
V
†
.
Since,
moreover,
X
η
†
→
X
η
∗
is
totally
wildly
ramified
over
the
terminal
vertices
of
γ
V
ν
∗
,
it
thus
follows
that
we
may
apply
Proposition
2.5,
(xii),
to
conclude
that
X
η
†
→
X
η
∗
is
wildly
ramified
at
every
non-stable
vertex
of
γ
V
ν
†
.
Write
B
for
the
set
of
irreducible
components
of
V
[which
we
think
of
as
valuations
of
the
function
field
of
X
η
×
k
k]
that
are
the
images
of
stable
vertices
of
γ
V
ν
†
,
for
nodes
ν
48
SHINICHI
MOCHIZUKI
of
X.
Observe
that
if
we
keep
the
coverings
X
η
†
→
X
η
∗
→
X
η
fixed,
but
vary
the
log-modification
V
log
→
X
log
[among,
say,
elements
of
M],
then
the
set
B
remains
unchanged
[if
we
think
of
B
as
a
set
of
valuations
of
the
function
field
of
X
η
×
k
k]
and
of
finite
cardinality
[bounded
by
the
cardinality
of
the
set
of
stable
irreducible
components
of
X
†
].
Thus,
in
summary,
if
we
think
of
B
as
a
subset
of
C,
then
we
may
conclude
the
following:
Λ
C
Λ
X
∗
surjects
onto
Λ
X
∗
/Λ
X
†
,
for
all
C
∈
C\B.
Since
B
is
finite,
it
thus
follows
that
there
exists
a
Λ-covering
X
η
→
X
η
†
→
X
η
such
that
Λ
C
Λ
X
†
has
nontrivial
image
in
Λ
X
†
/Λ
X
,
for
all
C
∈
C
[cf.
property
(3)].
This
completes
the
proof
of
property
(7).
Next,
let
us
consider
Λ-coverings
X
η
→
X
η
→
X
[which
exist
after
possibly
replacing
k
by
a
finite
extension
of
k]
such
that
Λ
X
/Λ
X
is
of
order
p,
and
Λ
C
has
nontrivial
image
in
Λ
X
/Λ
X
[which
implies
that
Λ
C
Λ
X
surjects
onto
Λ
X
/Λ
X
—
cf.
property
(3)]
for
all
C
∈
C
[cf.
property
(7)];
write
(X
)
log
→
(X
)
log
→
X
log
for
the
resulting
morphisms
of
log
stable
curves.
Let
(V
)
log
→
(X
)
log
,
(V
)
log
→
(X
)
log
be
split,
base-field-isomorphic,
unramified
log-modifications
such
that
there
exists
a
finite,
loop-preserving
morphism
(V
)
log
→
(V
)
log
lying
over
(X
)
log
→
(X
)
log
[which
exist
after
possibly
replacing
k
by
a
finite
extension
of
k
—
cf.
properties
(4),
(6);
Proposition
2.5,
(iii)].
Next,
let
us
consider
the
logarithmic
derivative
δ
:
ω
(V
)
log
/S
log
|
V
→
ω
(V
)
log
/S
log
of
the
morphism
(V
)
log
→
(V
)
log
.
Since
this
morphism
is
finite
étale
over
η,
it
follows
that
δ
is
an
isomorphism
over
η.
On
the
other
hand,
since
X
η
→
X
η
is
totally
wildly
ramified
over
every
irreducible
component
C
of
V
[i.e.,
induces
a
purely
inseparable
extension
of
degree
p
of
the
function
field
of
C],
it
follows
that
δ
vanishes
on
the
special
fiber
V
.
Write
δ
∗
=
π
−n
·
δ,
for
the
maximal
integer
n
such
that
π
−n
·
δ
remains
integral.
Thus,
we
obtain
a
morphism
def
δ
∗
:
ω
(V
)
log
/S
log
|
V
→
ω
(V
)
log
/S
log
of
line
bundles
on
V
which
is
not
identically
zero
on
V
.
Let
C
be
an
irreducible
component
of
V
such
that
δ
∗
|
C
≡
0.
Note
that
since
V
has
at
least
one
stable
irreducible
component,
it
follows
that
we
may
choose
C
such
that
either
C
is
stable
or
C
meets
an
irreducible
component
C
∗
of
V
such
that
δ
∗
|
C
∗
≡
0.
Thus,
if
C
is
not
stable,
then
it
follows
that
δ
∗
|
C
has
at
least
one
zero
[i.e.,
is
not
an
isomorphism
of
line
bundles].
Write
C
for
the
irreducible
component
of
V
which
is
the
image
of
C
;
E
→
C
,
E
→
C
for
the
respective
normalizations;
g
E
,
g
E
for
the
respective
genera
of
E
,
E
.
Also,
let
us
refer
to
the
points
of
C
,
E
,
C
,
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
49
E
that
do
not
map
to
the
respective
1-interiors
of
(V
)
log
,
(V
)
log
as
critical
points.
Write
D
E
⊆
E
,
D
E
⊆
E
for
the
respective
divisors
of
critical
points;
r
E
,
r
E
for
the
respective
degrees
of
D
E
,
D
E
.
Next,
let
us
consider
the
morphism
C
→
C
.
Since
this
morphism
C
→
C
induces
a
purely
inseparable
extension
of
degree
p
on
the
respective
function
fields,
∼
it
follows
that
we
have
an
isomorphism
of
k-schemes
E
×
k
k
→
E
[so
g
E
=
g
E
],
where
we
write
k
→
k
∼
=
k
for
the
[degree
one!]
field
extension
determined
by
the
Frobenius
morphism
on
k.
Next,
I
claim
that
the
critical
points
of
C
map
to
critical
points
of
C
.
Indeed,
if
a
critical
point
of
C
maps
to
a
non-critical
point
c
of
C
,
then
let
us
write
C
c
,
C
c
for
the
spectra
of
the
respective
completions
of
the
local
rings
of
C
,
C
at
[the
fiber
over]
c.
Then
observe
that
since
V
is
regular
[of
dimension
two]
at
c
[cf.
Proposition
2.5,
(iv)],
while
V
is
the
normalization
of
V
in
X
η
,
it
follows
from
elementary
commutative
algebra
that
V
is
finite
and
flat
over
V
of
degree
p
at
c.
Thus,
if
we
write
η
c
for
the
spectrum
of
the
residue
field
of
the
unique
generic
point
of
the
irreducible
scheme
C
c
,
then
C
c
×
C
c
η
c
→
η
c
is
finite,
flat
of
degree
≤
p;
[since
X
η
→
X
η
is
totally
wildly
ramified
over
every
irreducible
component
of
V
,
it
follows
that]
the
degree
of
each
of
the
≥
2
[cf.
Proposition
2.5,
(vi)]
connected
components
of
C
c
×
C
c
η
c
over
η
c
is
equal
to
p
—
in
contradiction
to
the
fact
that
the
degree
of
C
c
×
C
c
η
c
over
η
c
is
≤
p.
This
completes
the
proof
of
the
claim.
In
particular,
it
follows
that
r
E
≤
r
E
.
Now
recall
that
we
have
natural
isomorphisms
∼
ω
E
/k
(D
E
)
→
ω
(V
)
log
/S
log
|
E
;
∼
ω
E
/k
(D
E
)
→
ω
(V
)
log
/S
log
|
E
[cf.
Proposition
2.5,
(i),
(ii),
(iii),
(iv),
(v),
(vi);
the
fact
that
the
log-modifications
(V
)
log
→
(X
)
log
,
(V
)
log
→
(X
)
log
are
unramified].
Moreover,
it
follows
immedi-
ately
from
the
definitions
that
deg(ω
E
/k
(D
E
))
=
2g
E
+
r
E
,
deg(ω
E
/k
(D
E
))
=
2g
E
+r
E
.
We
thus
conclude
that
deg(ω
(V
)
log
/S
log
|
C
)
≤
deg(ω
(V
)
log
/S
log
|
C
).
On
the
other
hand,
the
existence
of
the
generically
nonzero
morphism
of
line
bundles
δ
∗
|
C
implies
that
deg(ω
(V
)
log
/S
log
|
C
)
≥
deg(ω
(V
)
log
/S
log
|
C
)
=
p
·
deg(ω
(V
)
log
/S
log
|
C
)
≥
p
·
deg(ω
(V
)
log
/S
log
|
C
)
—
which
implies
that
deg(ω
(V
)
log
/S
log
|
C
)
≤
0.
Now
if
C
is
stable,
then
we
have
deg(ω
(V
)
log
/S
log
|
C
)
=
2g
E
+
r
E
>
0.
We
thus
conclude
that
C
is
non-
stable.
But
this
implies
that
δ
∗
|
C
has
at
least
one
zero,
so
[cf.
the
above
display
of
inequalities]
we
obtain
that
deg(ω
(V
)
log
/S
log
|
C
)
<
0,
in
contradiction
to
the
equality
deg(ω
(V
)
log
/S
log
|
C
)
=
0
if
C
is
non-stable
[cf.
the
proof
of
Proposition
2.5,
(v)].
This
completes
the
proof
of
assertion
(ii).
In
light
of
assertion
(ii),
assertion
(iii)
follows
immediately
from
Proposition
2.5,
(viii).
This
completes
the
proof
of
assertion
(iii).
Next,
we
consider
assertion
(iv).
First,
let
us
observe
that
by
Proposition
2.5,
(ix),
it
follows
that
we
may
assume
that
split,
base-field-isomorphic
log-modifications
W
log
→
Y
log
,
Q
log
→
X
log
,
together
with
a
morphism
W
log
→
Q
log
over
Y
log
→
X
log
,
have
been
chosen
so
that
the
generic
point
of
the
unique
irreducible
compo-
[1]
nent
C
W
of
W
that
maps
finitely
to
C
maps
into
U
Q
.
Then
observe
that
there
are
precisely
two
mutually
exclusive
possibilities:
50
SHINICHI
MOCHIZUKI
[1]
(c)
some
nonempty
open
subscheme
of
C
maps
quasi-finitely
to
U
Q
;
[1]
(d)
C
maps
to
a
closed
point
c
of
U
Q
.
Moreover,
by
Proposition
2.5,
(i),
(ii),
(vii),
it
follows
immediately
that
(b)
[as
in
the
statement
of
assertion
(iv)]
⇐⇒
(d).
Thus,
it
suffices
to
show
that
(a)
[as
in
the
statement
of
assertion
(iv)]
⇐⇒
(d).
Note
that
it
is
immediate
that
(d)
implies
(a):
Indeed,
if
we
write
c
log
for
the
log
scheme
obtained
by
equipping
c
with
the
restriction
to
c
of
the
log
structure
of
Q
log
,
then
we
obtain
an
open
injection
π
1
(c
log
)
→
G
k
log
;
but
this
implies
that
the
natural
homomorphism
Δ
C
→
Δ
X
log
factors
through
{1}
=
Ker(π
1
(c
log
)
→
G
k
log
),
hence
that
condition
(a)
is
satisfied.
Thus,
it
remains
to
show
that
(a)
implies
(d),
or,
equivalently,
that
condition
(c)
implies
that
condition
(a)
fails
to
hold.
But
this
follows
immediately
from
the
observation
that
condition
(c)
implies
that
Δ
C
surjects
onto
an
open
subgroup
of
the
decomposition
group
Δ
E
in
Δ
X
log
of
some
irreducible
component
E
of
Q.
Here,
we
recall
that
the
following
well-known
facts:
if
E
is
stable,
then
Δ
E
may
be
identified
[1]
def
with
the
maximal
pro-l
quotient
of
π
1
(U
E
×
k
k),
where
we
write
U
E
=
E
U
Q
,
which
is
infinite;
if
E
is
not
stable,
then
Δ
E
∼
=
Z
l
(1)
[cf.
Proposition
2.5,
(v)],
hence
infinite.
This
completes
the
proof
of
assertion
(iv).
Finally,
we
consider
assertion
(v).
First,
we
observe
that
Δ
C
may
be
identified
[1]
def
with
the
maximal
pro-l
quotient
of
π
1
(U
C
×
k
k),
where
we
write
U
C
=
C
U
Y
[and
assume,
for
simplicity,
that
Y
log
is
split].
In
particular,
an
automorphism
of
U
C
is
equal
to
the
identity
if
and
only
if
it
induces
the
identity
outer
automorphism
of
Δ
C
[cf.,
e.g.,
[MT],
Proposition
1.4,
and
its
proof].
Note,
moreover,
that
an
automorphism
of
Y
log
stabilizes
C
if
and
only
if
it
stabilizes
the
conjugacy
class
of
Δ
C
[cf.,
e.g.,
[Mzk13],
Proposition
1.2,
(i)].
Thus,
assertion
(v)
reduces
to
the
[easily
verified]
assertion
that
the
morphism
Y
η
→
X
η
is
wildly
ramified
at
C
if
and
only
if
Gal(Y
η
/X
η
)
stabilizes
and
induces
the
identity
on
C.
This
completes
the
proof
of
assertion
(v).
Remark
2.6.1.
Note
that
the
content
of
Lemma
2.6,
(ii),
(iii),
is
reminiscent
of
the
main
results
of
[Tama2]
[cf.
also
Corollary
2.11
below].
By
comparison
to
Tamagawa’s
“resolution
of
nonsingularities”,
however,
Lemma
2.6,
(ii),
(iii),
assert
a
somewhat
weaker
conclusion,
albeit
for
pro-p
geometric
fundamental
groups,
as
opposed
to
profinite
geometric
fundamental
groups.
Remark
2.6.2.
The
argument
applied
in
the
final
portion
of
the
proof
of
Lemma
2.6,
(ii),
is
reminiscent
of
the
well-known
classical
argument
that
implies
the
nonex-
istence
of
a
Frobenius
lifting
for
stable
curves
over
the
ring
of
Witt
vectors
of
a
finite
field.
That
is
to
say,
if
k
is
absolutely
unramified,
and
Φ:
X
→
X
is
an
S-morphism
that
induces
the
Frobenius
morphism
between
the
respective
spe-
cial
fibers,
then
one
obtains
a
contradiction
as
follows:
Since
Φ
induces
a
morphism
X
η
→
X
η
,
it
follows
immediately
that
Φ
extends
to
a
morphism
of
log
stable
curves
Φ
log
:
X
log
→
X
log
.
Although
the
derivative
dΦ
log
:
Φ
∗
(ω
X
log
/S
log
)
→
ω
X
log
/S
log
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
51
is
≡
0
(mod
p),
one
verifies
immediately
[by
an
easy
local
calculation]
that
p
1
dΦ
log
is
necessarily
≡
0
(mod
p)
generically
on
each
irreducible
component
of
X.
Since
ω
X
log
/S
log
is
a
line
bundle
of
degree
2g
X
−
2,
and
Φ
reduces
to
the
Frobenius
mor-
phism
between
the
special
fibers,
the
existence
of
dΦ
log
thus
implies
[by
taking
degrees]
that
p
·
(2g
X
−
2)
≤
2g
X
−
2,
i.e.,
that
(p
−
1)(2g
X
−
2)
≤
0,
in
contradic-
tion
to
the
fact
that
g
X
≥
2.
Remark
2.6.3.
Note
that
it
follows
immediately
from
either
of
the
conditions
(a),
(b)
of
Lemma
2.6,
(ii),
that
Y
is
not
k-smooth
[i.e.,
“singular”].
Corollary
2.7.
(Uniformly
Toral
Neighborhoods
via
Cyclic
Coverings)
Suppose
that
we
are
either
in
the
situation
of
Lemma
2.6,
(ii),
(a)
—
which
we
shall
refer
to
in
the
following
as
case
(a)
—
or
in
the
situation
of
Lemma
2.6,
(ii),
(b)
—
which
we
shall
refer
to
in
the
following
as
case
(b);
suppose
further,
in
case
(b),
that
X
is
not
smooth
over
k.
Also,
we
suppose
that
we
have
been
given
a
commutative
diagram
as
in
Lemma
2.6,
(iii).
Thus,
in
either
case,
we
have
an
irreducible
component
C
W
of
W
[lying
over
an
irreducible
component
C
of
Y
]
[1]
satisfying
certain
special
properties,
as
in
Lemma
2.6,
(iii).
Let
y
∈
U
W
(O
k
)
(⊆
W
η
(k)
=
Y
η
(k))
be
a
point
such
that
the
image
of
y
meets
C
W
,
and,
moreover,
y
[1]
maps
to
a
point
x
∈
U
Q
(O
k
)
(⊆
Q
η
(k)
=
X
η
(k));
C
Q
the
irreducible
component
of
Q
that
meets
the
image
of
x;
def
F
y
=
W
\(C
W
[1]
U
W
)
⊆
W
;
def
F
x
=
Q\(C
Q
[1]
U
Q
)
⊆
Q
[regarded
as
closed
subsets
of
W
,
Q];
def
U
y
=
W
\F
y
⊆
W
;
def
U
x
=
Q\F
x
⊆
Q
[so
the
image
of
y
lies
in
U
y
;
the
image
of
x
lies
in
U
x
].
Write
g
Y
for
the
genus
of
Y
η
;
J
η
Y
→
η
(respectively,
J
η
X
→
η)
for
the
Jacobian
of
Y
η
(respectively,
X
η
);
J
Y
→
S
(respectively,
J
X
→
S)
for
the
uniquely
determined
semi-abelian
scheme
over
S
that
extends
J
η
Y
(respectively,
J
η
X
);
ι
Yη
:
Y
η
→
J
η
Y
for
the
morphism
that
sends
a
T
-valued
point
ζ
[where
T
is
a
k-scheme]
of
Y
η
,
regarded
as
a
divisor
on
Y
η
×
k
T
,
to
the
point
of
J
η
Y
determined
by
the
degree
zero
divisor
ζ
−
(y|
T
).
In
case
(a),
let
σ
∈
Gal(Y
η
/X
η
)
be
a
generator
of
Gal(Y
η
/X
η
);
J
η
⊆
J
η
Y
the
image
abelian
scheme
of
the
restriction
to
η
of
the
endomorphism
(1
−
σ)
:
J
Y
→
J
Y
;
J
→
S
the
uniquely
determined
semi-abelian
scheme
over
S
that
extends
J
η
[which
exists,
for
instance,
by
[BLR],
§7.4,
Lemma
2];
κ
:
J
Y
→
J
the
[dominant]
morphism
induced
by
(1
−
σ).
In
case
(b),
let
J
→
S
be
the
semi-
abelian
scheme
J
X
→
S;
κ
:
J
Y
→
J
the
[dominant]
morphism
induced
by
the
covering
Y
η
→
X
η
.
Write
β
η
:
Y
η
×
k
.
.
.
×
k
Y
η
→
J
η
Y
→
J
η
52
SHINICHI
MOCHIZUKI
[where
the
product
is
of
g
Y
copies
of
Y
η
]
for
the
composite
of
the
morphism
given
def
by
adding
g
Y
copies
of
ι
Yη
with
the
morphism
κ
η
=
κ|
η
.
m
)
for
the
formal
group
over
S
given
by
com-
(i)
Write
J
(respectively,
G
pleting
J
(respectively,
the
multiplicative
group
(G
m
)
S
over
S)
at
the
origin.
Then
there
exists
an
exact
sequence
0
→
J
→
J
→
J
→
0
∼
of
[formally
smooth]
formal
groups
over
S,
together
with
an
isomorphism
J
→
G
m
∼
of
formal
groups
over
S.
In
the
following,
let
us
fix
such
an
isomorphism
J
→
G
m
and
identify
J
with
its
image
in
J.
(ii)
The
morphisms
ι
Yη
,
β
η
extend
uniquely
to
morphisms
ι
Y
:
U
y
→
J
Y
;
β
:
U
y
×
k
.
.
.
×
k
U
y
→
J
[where
the
product
is
of
g
Y
copies
of
U
y
],
respectively;
the
morphism
W
→
Q
restricts
to
a
morphism
U
y
→
U
x
.
(iii)
Suppose
that
k
is
an
MLF
[or,
equivalently,
that
k
is
finite].
Then
there
exists
a
positive
integer
M
—
which,
in
fact,
may
be
taken
to
be
1
in
case
(b)
—
such
that
the
following
condition
holds:
Let
k
•
⊆
k
be
a
finite
extension
of
k
with
ring
of
integers
O
k
•
,
maximal
ideal
m
k
•
⊆
O
k
•
.
Write
I
k
•
for
the
image
in
J(O
k
•
)
via
β
[cf.
(ii)]
of
the
product
of
g
Y
copies
of
U
y
(O
k
•
).
Then
M
·
I
k
•
lies
in
the
k
)
⊆
J(O
k
).
Write
I
k
⊆
J(O
k
)
for
the
subgroup
determined
by
subgroup
J(O
•
•
•
•
M
·
I
k
•
;
∼
∼
N
k
•
⊆
(O
k
×
•
)
⊗
Q
p
→
G
m
(O
k
•
)
⊗
Q
p
→
J
(O
k
•
)
⊗
Q
p
for
the
image
of
the
intersection
J
(O
k
•
)
I
k
•
k
))
(⊆
J(O
•
in
J
(O
k
•
)⊗Q
p
.
Then
as
k
•
⊆
k
varies
over
the
finite
extensions
of
k,
the
subgroups
N
k
•
determine
a
uniformly
toral
neighborhood
of
Gal(k/k)
[cf.
[Mzk15],
Def-
inition
3.6,
(i),
(ii)].
Proof.
First,
we
consider
assertion
(i).
Recall
from
the
well-known
theory
of
Néron
models
of
Jacobians
[cf.,
e.g.,
[BLR],
§9.2,
Example
8]
that
the
torus
portion
of
the
special
fiber
of
J
Y
(respectively,
J
X
)
is
[in
the
notation
of
Proposition
2.5,
(vi)]
of
rank
lp-rk(Y
)
(respectively,
lp-rk(X)).
In
particular,
the
torus
portion
of
the
special
fiber
of
J
is
of
rank
lp-rk(Y
)
−
lp-rk(X)
in
case
(a),
and
of
rank
lp-rk(X)
in
case
(b).
Thus,
in
case
(a),
the
fact
that
the
morphism
Y
η
→
X
η
is
loopifying
implies
that
the
torus
portion
of
the
special
fiber
of
J
is
of
positive
rank;
in
case
(b),
since
X
is
not
k-smooth,
it
follows
from
the
loop-ampleness
assumption
of
Lemma
2.6
that
lp-rk(X)
>
0,
hence
that
the
torus
portion
of
the
special
fiber
of
J
is
of
positive
rank.
Now
the
existence
of
an
exact
sequence
as
in
assertion
(i)
follows
from
the
well-known
theory
of
degeneration
of
abelian
varieties
[cf.,
e.g.,
[FC],
Chapter
III,
Corollary
7.3].
This
completes
the
proof
of
assertion
(i).
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
53
Next,
we
consider
assertion
(ii).
The
existence
of
the
unique
extension
of
ι
Yη
follows
immediately
from
Proposition
2.5,
(x);
the
existence
of
the
unique
extension
of
β
η
follows
immediately
from
the
existence
of
this
unique
extension
of
ι
Yη
[together
with
the
existence
of
the
homomorphism
of
semi-abelian
schemes
κ
:
J
Y
→
J].
In
case
(b),
the
existence
of
the
morphism
U
y
→
U
x
follows
immediately
from
the
definitions.
In
case
(a),
if
the
morphism
U
y
→
U
x
fails
to
exist,
then
there
exists
a
[1]
closed
point
w
∈
W
that
maps
to
a
closed
point
q
∈
Q
such
that
w
∈
U
y
⊆
U
W
,
but
[1]
q
∈
U
Q
.
On
the
other
hand,
since
there
exists
an
irreducible
Zariski
neighborhood
of
w
in
W
[cf.
the
simple
structure
of
the
monoid
N],
it
follows
from
the
fact
that
C
W
maps
finitely
to
C
Q
[in
case
(a)],
that
W
→
Q
is
quasi-finite
in
a
Zariski
neighborhood
of
w.
Thus,
if
we
write
R
w
,
R
q
for
the
respective
strict
henselizations
of
W
,
Q
at
[some
choice
of
k-valued
points
lifting]
w,
q,
then
R
w
,
R
q
are
normal,
and
theorem].
In
particular,
the
natural
inclusion
R
q
→
R
w
is
finite
[cf.
Zariski’s
main
×
K
q
=
R
q
×
[where
“×”
if
we
write
K
q
for
the
quotient
field
of
R
q
,
then
we
have
R
w
denotes
the
subgroup
of
units],
so
the
morphism
W
log
→
Q
log
induces
an
injection
on
the
groupifications
of
the
stalks
of
the
characteristic
sheaves
at
[some
choice
[1]
of
k-valued
points
lifting]
w,
q
—
in
contradiction
to
the
fact
that
w
∈
U
W
,
but
[1]
q
∈
U
Q
.
This
completes
the
proof
of
assertion
(ii).
Finally,
we
consider
assertion
(iii).
First,
we
define
the
number
M
as
follows:
In
case
(a),
the
endomorphism
(1−σ)
:
J
Y
→
J
Y
admits
a
factorization
θ◦κ,
where
θ
:
J
→
J
Y
is
a
“closed
immersion
up
to
isogeny”
[cf.,
e.g.,
the
situation
discussed
in
[BLR],
§7.5,
Proposition
3,
(b)]
—
i.e.,
there
exists
a
morphism
θ
:
J
Y
→
J
such
that
θ
◦
θ
:
J
→
J
is
multiplication
by
some
positive
integer
M
θ
on
J;
then
def
def
we
take
M
=
M
θ
.
In
case
(b),
we
take
M
=
1.
In
the
following,
if
G
is
a
group
scheme
or
formal
group
over
O
k
,
and
r
≥
1
is
an
integer,
then
let
us
write
G
m
r
(O
k
•
)
⊆
G(O
k
•
)
for
the
subgroup
of
elements
that
are
congruent
to
the
identity
modulo
m
rk
·
O
k
•
.
Next,
let
us
make
the
following
observation:
(1)
We
have
M
·
I
k
•
⊆
J
m
(O
k
•
)
⊆
J(O
k
•
).
Indeed,
in
case
(a),
we
reason
as
follows:
It
suffices
to
show
that
M
·κ(ι
Y
(U
y
(O
k
•
)))
⊆
J
m
(O
k
•
).
Since,
moreover,
the
endomorphism
(1
−
σ)
:
J
Y
→
J
Y
admits
a
factor-
ization
θ
◦
κ,
where,
for
some
morphism
θ
:
J
Y
→
J,
θ
◦
θ
is
equal
to
multiplication
by
M
,
it
suffices
to
show
that
Y
(O
k
•
)
(1
−
σ)(ι
Y
(U
y
(O
k
•
)))
⊆
J
m
[since
applying
θ
to
this
inclusion
yields
the
desired
inclusion
M
·κ(ι
Y
(U
y
(O
k
•
)))
⊆
J
m
(O
k
•
)].
On
the
other
hand,
since
Y
η
→
X
η
is
wildly
ramified
at
C
W
,
it
follows
that
σ
acts
as
the
identity
on
C
W
,
hence
that
the
composite
morphism
(1
−
σ)
◦
ι
Y
:
U
y
→
J
Y
induces
a
morphism
on
special
fibers
U
y
×
O
k
k
→
J
Y
×
O
k
k
that
is
constant
[with
image
lying
in
the
image
of
the
identity
section
of
J
Y
×
O
k
k].
But
this
implies
Y
(O
k
•
).
This
completes
the
proof
of
observation
(1)
that
(1
−
σ)(ι
Y
(U
y
(O
k
•
)))
⊆
J
m
in
case
(a).
In
a
similar
[but
slightly
simpler]
vein,
in
case
(b),
it
suffices
to
observe
54
SHINICHI
MOCHIZUKI
that
the
morphism
κ
◦
ι
Y
:
U
y
→
J
admits
a
factorization
U
y
→
U
x
→
J
X
,
where
U
y
→
U
x
is
the
morphism
of
assertion
(ii),
and
U
x
→
J
X
is
the
“analogue
of
ι
Y
”
for
the
point
x
of
X
η
(k)
[cf.
Proposition
2.5,
(x)].
That
is
to
say,
the
fact
that
C
W
[1]
maps
to
x
∈
U
Q
(O
k
)
implies
[by
applying
this
factorization]
that
the
morphism
κ
◦
ι
Y
:
U
y
→
J
induces
a
morphism
on
special
fibers
U
y
×
O
k
k
→
J
×
O
k
k
that
is
constant
[with
image
lying
in
the
image
of
the
identity
section
of
J
×
O
k
k].
This
completes
the
proof
of
observation
(1)
in
case
(b).
Next,
let
us
make
the
following
observation:
(2)
There
exists
a
positive
integer
r
which
is
independent
of
k
•
such
that
M
·
I
k
•
⊇
J
m
r
(O
k
•
).
Indeed,
since
κ
is
clearly
dominant,
it
follows
immediately
that
the
composite
of
the
morphism
β
:
U
y
×
k
.
.
.
×
k
U
y
→
J
with
the
morphism
J
→
J
given
by
multiplication
by
M
is
dominant,
hence,
in
particular,
generically
smooth
[since
k
is
of
characteristic
zero].
Thus,
[since
M
·
I
k
•
is
a
group!]
observation
(2)
follows
immediately
from
the
“positive
slope
version
of
Hensel’s
lemma”
given
in
Lemma
(O
k
•
)
=
J
m
(O
k
•
)
J
(O
k
•
)
[cf.
assertion
(i)],
we
conclude
that
2.1.
Now
since
J
m
J
m
r
(O
k
•
)
⊆
I
k
•
J
(O
k
•
)
⊆
J
m
(O
k
•
)
[cf.
the
inclusions
of
observations
(1),
(2)],
so
assertion
(iii)
follows
essentially
formally
[cf.
[Mzk15],
Definition
3.6,
(i),
(ii)].
This
completes
the
proof
of
assertion
(iii).
Remark
2.7.1.
Note
that
in
the
situation
of
case
(b),
if
f
is
a
rational
function
on
X
whose
value
at
x
lies
in
O
k
×
,
then
the
values
∈
O
k
×
•
of
f
at
points
of
U
y
(O
k
•
)
[cf.
the
notation
of
Corollary
2.7,
(iii)]
determine
a
uniformly
toral
neighborhood.
It
was
precisely
this
observation
that
motivated
the
author
to
develop
the
theory
of
the
present
§2.
Definition
2.8.
k.
Let
k
be
a
field
of
characteristic
zero,
k
an
algebraic
closure
of
(i)
Suppose
that
k
is
equipped
with
a
topology.
Let
X
be
a
smooth,
geo-
metrically
connected
curve
over
k.
Then
we
shall
say
that
a
subset
Ξ
⊆
X(k)
is
Galois-dense
if,
for
every
finite
extension
field
k
⊆
k
of
k,
Ξ
X(k
)
is
dense
in
X(k
)
[i.e.,
relative
to
the
topology
induced
on
X(k
)
by
k].
(ii)
We
shall
refer
to
as
a
pro-curve
U
over
k
[cf.
the
terminology
of
[Mzk3]]
any
k-scheme
U
that
may
be
written
as
a
projective
limit
of
smooth,
geometrically
connected
curves
over
k
in
which
the
transition
morphisms
are
birational.
Let
U
be
a
pro-curve
over
k.
Then
it
makes
sense
to
speak
of
the
function
field
k(U
)
of
U
.
Write
X
for
the
smooth,
proper,
geometrically
connected
curve
over
k
determined
by
the
function
field
k(U
).
Then
one
verifies
immediately
that
U
is
completely
determined
up
to
unique
isomorphism
by
k(U
),
together
with
some
Gal(k/k)-stable
subset
Ξ
⊆
X(k)
—
i.e.,
roughly
speaking,
“U
is
obtained
by
removing
Ξ
from
X”.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
55
If
k
is
equipped
with
a
topology,
then
we
shall
say
that
U
is
co-Galois-dense
if
the
corresponding
Gal(k/k)-stable
subset
Ξ
⊆
X(k)
is
Galois-dense.
Remark
2.8.1.
Suppose,
in
the
notation
of
Definition
2.8,
that
k
is
an
MLF
[and
that
k
is
equipped
with
the
p-adic
topology].
Let
X
be
a
smooth,
proper,
geometrically
connected
curve
over
k,
with
function
field
k(X).
Then
Spec(k(X))
is
a
co-Galois-dense
pro-curve
over
k.
Suppose
that
X
=
X
0
×
k
0
k,
where
k
0
⊆
k
is
a
number
field,
and
X
0
is
a
smooth,
proper,
geometrically
connected
curve
over
k
0
,
with
function
field
k(X
0
).
Then
Spec(k(X
0
)
⊗
k
0
k)
[where
we
note
that
the
ring
k(X
0
)
⊗
k
0
k
is
not
a
field!]
also
forms
an
example
of
a
co-Galois-dense
pro-curve
over
k.
Remark
2.8.2.
Let
k
be
a
field
of
characteristic
zero.
(i)
Let
us
say
that
a
pro-curve
U
over
k
is
of
unit
type
if
there
exists
a
connected
finite
étale
covering
of
U
that
admits
a
nonconstant
unit.
Thus,
any
hyperbolic
curve
U
over
k
for
which
there
exists
a
connected
finite
étale
covering
V
→
U
such
that
V
admits
a
dominant
k-morphism
V
→
P
,
where
P
is
the
projective
line
minus
three
points
over
k,
is
of
unit
type.
That
is
to
say,
the
hyperbolic
curves
considered
in
[Mzk15],
Remark
3.8.1
—
i.e.,
the
sort
of
hyperbolic
curves
that
motivated
the
author
to
prove
[Mzk15],
Corollary
3.8,
(g)
—
are
necessarily
of
unit
type.
(ii)
Suppose
that
k
is
an
MLF
of
residue
characteristic
p,
whose
ring
of
integers
we
denote
by
O
k
.
Let
n
≥
1
be
an
integer;
η
∈
O
k
/(p
n
).
Then
observe
that
the
set
E
of
elements
of
O
k
that
are
≡
η
(mod
p
n
)
is
of
uncountable
cardinality.
In
particular,
it
follows
that
the
subfield
of
k
generated
over
Q
by
E
is
of
uncountable
—
hence,
in
particular,
infinite
—
transcendence
degree
over
Q.
(iii)
Let
k
be
as
in
(ii);
X
0
a
proper
hyperbolic
curve
over
k
0
,
where
k
0
⊆
k
is
a
finitely
generated
extension
of
Q;
k
1
⊆
k
a
finitely
generated
extension
of
k
0
;
r
≥
1
an
integer.
Then
recall
from
[MT],
Corollary
5.7,
that
any
curve
U
1
obtained
by
def
removing
from
X
1
=
X
0
×
k
0
k
1
a
set
of
r
“generic
points”
∈
X
1
(k
1
)
=
X
0
(k
1
)
—
i.e.,
r
points
which
determine
a
dominant
morphism
from
Spec(k
1
)
to
the
product
of
r
copies
of
X
0
over
k
0
—
is
not
of
unit-type.
In
particular,
it
follows
immediately
from
(ii)
that:
There
exist
co-Galois-dense
pro-curves
U
over
k
which
are
not
of
unit
type.
For
more
on
the
significance
of
this
fact,
we
refer
to
Remark
2.10.1
below.
Remark
2.8.3.
Suppose
that
we
are
in
the
situation
of
Definition
2.8,
(i).
Let
Y
→
X
be
a
connected
finite
étale
covering.
Then
one
verifies
immediately,
by
applying
“Krasner’s
Lemma”
[cf.
[Kobl],
pp.
69-70],
that
the
inverse
image
Ξ|
Y
⊆
Y
(k)
of
a
Galois-dense
subset
Ξ
⊆
X(k)
is
itself
Galois-dense.
56
SHINICHI
MOCHIZUKI
Corollary
2.9.
(Point-theoreticity
Implies
Geometricity)
For
i
=
1,
2,
let
k
i
be
an
MLF
of
residue
characteristic
p
i
;
k
i
an
algebraic
closure
of
k
i
;
Σ
i
a
set
of
primes
of
cardinality
≥
2
such
that
p
i
∈
Σ
i
;
X
i
a
hyperbolic
curve
over
k
i
;
X
i
the
smooth,
proper,
geometrically
connected
curve
over
k
i
determined
by
the
function
field
of
X
i
;
Ξ
i
⊆
X
i
(k
i
)
a
Galois-dense
subset.
Write
“π
1
(−)”
for
the
étale
fundamental
group
of
a
connected
scheme,
relative
to
an
appropriate
choice
of
basepoint;
Δ
X
i
for
the
maximal
pro-Σ
i
quotient
of
π
1
(X
i
×
k
i
k
i
);
Π
X
i
for
the
quotient
of
π
1
(X
i
)
by
the
kernel
of
the
natural
surjection
π
1
(X
i
×
k
i
k
i
)
Δ
X
i
.
Let
∼
α
:
Π
X
1
→
Π
X
2
be
an
isomorphism
of
profinite
groups
such
that
a
closed
subgroup
of
Π
X
1
is
a
decomposition
group
of
a
point
∈
Ξ
1
if
and
only
if
it
corresponds,
relative
to
α,
to
a
decomposition
group
in
Π
X
2
of
a
point
∈
Ξ
2
.
Then
p
1
=
p
2
,
Σ
1
=
Σ
2
,
and
α
is
∼
geometric,
i.e.,
arises
from
a
unique
isomorphism
of
schemes
X
1
→
X
2
.
Proof.
First,
we
observe
that
by
[Mzk15],
Theorem
2.14,
(i),
α
induces
iso-
∼
∼
morphisms
α
Δ
:
Δ
X
1
→
Δ
X
2
,
α
G
:
G
1
→
G
2
[where,
for
i
=
1,
2,
we
write
def
def
G
i
=
Gal(k
i
/k
i
)];
p
1
=
p
2
[so
we
shall
write
p
=
p
1
=
p
2
];
Σ
1
=
Σ
2
[so
we
shall
def
write
Σ
=
Σ
1
=
Σ
2
].
Also,
by
[the
portion
concerning
semi-graphs
of]
[Mzk15],
Theorem
2.14,
(i),
it
follows
that
α
preserves
the
decomposition
groups
of
cusps.
Thus,
by
passing
to
corresponding
open
subgroups
of
Π
X
i
[cf.
Remark
2.8.3]
and
forming
the
quotient
by
the
decomposition
groups
of
cusps
in
Δ
X
i
,
we
may
assume,
without
loss
of
generality,
that
the
X
i
are
proper.
Next,
let
l
∈
Σ
be
a
prime
=
p.
Then
let
us
recall
that
by
the
well-known
stable
reduction
criterion
[cf.,
e.g.,
[BLR],
§7.4,
Theorem
6],
X
i
has
stable
reduction
over
O
k
i
if
and
only
if,
for
some
Z
l
-
ab
submodule
M
⊆
Δ
ab
X
i
⊗
Z
l
of
the
maximal
pro-l
abelian
quotient
Δ
X
i
⊗
Z
l
of
Δ
X
i
,
the
inertia
subgroup
of
G
i
acts
trivially
on
M
,
Δ
ab
X
i
⊗
Z
l
/M
.
Thus,
we
may
assume,
without
loss
of
generality,
that,
for
i
=
1,
2,
X
i
admits
a
log
stable
model
X
i
log
over
Spec(O
k
i
)
log
[where
the
last
log
structure
is
the
log
structure
determined
by
the
closed
point].
Since,
by
[the
portion
concerning
semi-graphs
of]
[Mzk15],
Theorem
2.14,
(i),
it
follows
that
α
induces
an
isomorphism
between
the
dual
graphs
of
the
special
fibers
X
i
of
the
X
i
,
hence
that
X
1
is
loop-ample
(respectively,
singular)
if
and
only
if
X
2
is.
Thus,
by
replacing
X
i
by
a
finite
étale
covering
of
X
i
arising
from
an
open
subgroup
of
Π
X
i
,
we
may
assume
that
X
i
is
loop-ample
[cf.
§0]
and
singular
[cf.
Remark
2.6.3].
Now,
to
complete
the
proof
of
Corollary
2.9,
it
follows
from
[Mzk15],
Corollary
3.8,
(e),
that
it
suffices
to
show
that
α
G
is
uniformly
toral.
Next,
let
us
suppose
that,
for
i
=
1,
2,
we
are
given
a
finite
étale
covering
Y
i
→
X
i
of
hyperbolic
curves
over
k
i
with
stable
reduction
over
O
k
i
arising
from
open
subgroups
of
Π
X
i
that
correspond
via
α
and
are
such
that
Gal(Y
i
/X
i
)
is
cyclic
of
order
a
positive
power
of
p
[cf.
Lemma
2.6,
(ii)].
Let
us
write
Y
i
log
for
the
log
stable
model
of
Y
i
over
(O
k
i
)
log
,
Y
i
for
the
special
fiber
of
Y
i
.
By
possibly
replacing
the
k
i
by
corresponding
[relative
to
α]
finite
extensions
of
k
i
,
we
may
assume
that
the
Y
i
are
split
[cf.
Proposition
2.5,
(vi)].
Note
that
by
[the
portion
concerning
semi-graphs
of]
[Mzk15],
Theorem
2.14,
(i),
it
follows
that
Y
1
→
X
1
is
loopifying
if
and
only
if
Y
2
→
X
2
is.
Thus,
by
Lemma
2.6,
(iv),
(v)
[cf.
also
Proposition
2.5,
(i),
(ii)]
(respectively,
Lemma
2.6,
(iv)),
it
follows
that
Y
1
→
X
1
satisfies
condition
(a)
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
57
(respectively,
(b))
of
Lemma
2.6,
(ii),
if
and
if
Y
2
→
X
2
does.
For
i
=
1,
2,
let
C
i
be
an
irreducible
component
of
Y
i
as
in
Corollary
2.7
[i.e.,
“C”].
By
Lemma
2.6,
(iv)
[cf.
also
the
portion
concerning
semi-graphs
of
[Mzk15],
Theorem
2.14,
(i)],
we
may
assume
that
the
C
i
are
compatible
with
α.
Thus,
to
complete
the
proof
of
Corollary
2.9,
it
suffices
to
construct
uniformly
toral
neighborhoods
[cf.
Corollary
2.7,
(iii)]
that
are
compatible
with
α.
Write
Π
Y
i
⊆
Π
X
i
,
Δ
Y
i
⊆
Δ
X
i
for
the
open
subgroups
determined
by
Y
i
;
T
i
Y
,
T
i
X
for
the
maximal
pro-p
abelian
quotients
of
Δ
Y
i
,
Δ
X
i
.
If
we
are
in
case
(a)
[cf.
Corollary
2.7],
then
we
choose
generators
σ
i
∈
Gal(Y
i
/X
i
)
∼
=
Δ
X
i
/Δ
Y
i
that
correspond
via
α
and
write
T
i
for
the
intersection
with
T
i
Y
of
the
image
of
the
endomorphism
(1
−
σ
i
)
of
T
i
Y
⊗
Q
p
,
and
κ
T
i
:
T
i
Y
→
T
i
for
the
morphism
induced
by
(1
−
σ
i
).
If
we
are
in
case
(b)
[cf.
Corollary
2.7],
then
def
we
set
T
i
=
T
i
X
;
write
κ
T
i
:
T
i
Y
→
T
i
for
the
morphism
induced
by
Y
i
→
X
i
[i.e.,
by
the
inclusion
Δ
Y
i
→
Δ
X
i
].
Thus,
the
formal
group
“J
”
of
Corollary
2.7,
(i),
corresponds
to
a
G
i
-submodule
T
i
⊆
T
i
such
that
T
i
∼
=
Z
p
(1);
by
[Tate],
Theorem
4
[cf.
also
[Mzk5],
Proposition
1.2.1,
(vi)],
we
may
assume
that
these
submodules
T
i
are
compatible
with
α.
(l)
(l)
(l)
Next,
let
us
write
Δ
Y
i
Δ
Y
i
for
the
maximal
pro-l
quotient
of
Δ
Y
i
;
Δ
C
i
⊆
Δ
Y
i
(l)
for
the
decomposition
group
of
C
i
in
Δ
Y
i
[well-defined
up
to
conjugation].
Thus,
(l)
Δ
C
i
may
be
identified
with
the
maximal
pro-l
quotient
of
π
1
(U
C
i
×
k
i
k
i
),
where
[1]
def
U
C
i
=
C
i
U
Y
i
[cf.
the
proof
of
Lemma
2.6,
(iv)],
k
i
is
the
residue
field
of
k
i
,
(l)
and
k
i
is
the
algebraic
closure
of
k
i
induced
by
k
i
.
Since
Δ
C
i
is
slim
[cf.,
e.g.,
(l)
[MT],
Proposition
1.4],
and
the
outer
action
of
G
i
on
Δ
C
i
clearly
factors
through
(l)
the
quotient
G
i
Gal(k
i
/k
i
),
the
resulting
outer
action
of
Gal(k
i
/k
i
)
on
Δ
C
i
determines,
in
a
fashion
that
is
compatible
with
α,
an
extension
of
profinite
groups
(l)
(l)
1
→
Δ
C
i
→
Π
C
i
→
Gal(k
i
/k
i
)
→
1.
In
a
similar
vein,
the
outer
action
of
G
i
(l)
on
Δ
Y
i
factors
through
the
maximal
tamely
ramified
quotient
G
i
G
k
log
,
hence
i
(l)
[since
Δ
Y
i
is
slim
—
cf.,
e.g.,
[MT],
Proposition
1.4]
determines,
in
a
fashion
that
is
compatible
with
α,
a
morphism
of
extensions
of
profinite
groups
(l)
−→
Π
C
i
×
Gal(k
/k
)
G
k
log
i
⏐
i
i
⏐
(l)
−→
1
−→
Δ
C
i
⏐
⏐
1
−→
Δ
Y
i
(l)
(l)
Π
Y
i
−→
G
k
log
⏐
i
⏐
−→
1
−→
G
k
log
−→
1
i
—
in
which
the
vertical
morphisms
are
inclusions,
and
the
vertical
morphism
on
the
right
is
the
identity
morphism;
moreover,
the
images
of
the
first
two
vertical
morphisms
are
equal
to
the
respective
decomposition
groups
of
C
i
[well-defined
up
to
conjugation].
Next,
let
us
observe
that,
by
our
assumption
concerning
decomposition
groups
of
points
∈
Ξ
i
in
the
statement
of
Corollary
2.9,
it
follows
that
α
determines
∼
a
bijection
Y
1
(k
1
,
Ξ
1
)
→
Y
2
(k
2
,
Ξ
2
),
where
we
write
Y
i
(k
i
,
Ξ
i
)
⊆
Y
i
(k
i
)
for
the
58
SHINICHI
MOCHIZUKI
subset
of
points
lying
over
points
∈
Ξ
i
.
[Here,
we
recall
that
a
point
∈
Y
i
(k
i
)
is
uniquely
determined
by
the
conjugacy
class
of
its
decomposition
group
in
Π
Y
i
—
cf.,
e.g.,
[Mzk3],
Theorem
C.]
Now
let
us
choose
corresponding
[i.e.,
via
this
bijection]
points
y
i
∈
Y
i
(k
i
,
Ξ
i
)
as
our
points
“y”
in
the
construction
of
the
uniformly
toral
neighborhoods
of
Corollary
2.7,
(iii).
Here,
we
observe
that
[by
our
Galois-
density
assumption]
we
may
assume
that
y
i
is
compatible
with
C
i
,
in
the
sense
(l)
that
the
image
in
the
quotient
Π
Y
i
Π
Y
i
of
the
decomposition
group
of
y
i
in
(l)
Π
Y
i
determines
a
subgroup
of
Π
C
i
×
Gal(k
/k
)
G
k
log
which
contains
the
kernel
of
i
(l)
(l)
i
i
the
surjection
Π
C
i
×
Gal(k
/k
)
G
k
log
Π
C
i
.
Note
that
this
condition
that
y
i
be
i
i
i
“compatible
with
C
i
”
is
manifestly
“group-theoretic”,
i.e.,
compatible
with
α
[cf.
the
portion
concerning
semi-graphs
of
[Mzk15],
Theorem
2.14,
(i);
[Mzk5],
Proposition
1.2.1,
(ii)].
Moreover,
let
us
recall
from
the
theory
of
§1
that
this
condition
that
y
i
be
“compatible
with
C
i
”
is
equivalent
to
the
condition
that
the
closure
in
Y
i
of
y
i
intersect
U
C
i
[cf.
Proposition
1.3,
(x)].
Thus,
by
choosing
any
corresponding
[i.e.,
via
the
bijection
induced
by
α]
points
y
i
∈
Y
i
(k
i
,
Ξ
i
)
that
are
compatible
with
the
C
i
,
we
may
compute
[directly
from
the
decomposition
groups
of
the
y
i
,
y
i
in
Π
Y
i
]
the
“difference”
of
y
i
,
y
i
in
H
1
(G
k
i
,
T
i
Y
),
as
well
as
the
image
δ
y
i
,y
i
∈
H
1
(G
k
i
,
T
i
)
of
this
difference
via
κ
T
i
.
On
the
other
hand,
let
us
recall
the
Kummer
isomorphisms
H
1
(G
k
i
,
T
i
)
∼
=
O
k
×
i
⊗
Z
p
;
H
1
(G
k
i
,
T
i
)
∼
=
J
i
(k
i
)
⊗
Z
p
[where
J
i
is
the
“J”
of
Corollary
2.7,
(iii)
—
cf.,
e.g.,
the
“well-known
general
nonsense”
reviewed
in
the
proof
of
[Mzk14],
Proposition
2.2,
(i),
for
more
details].
By
applying
these
isomorphisms,
we
conclude
that
the
subset
of
H
1
(G
k
i
,
T
i
⊗
Q
p
)
∼
=
O
k
×
i
⊗
Q
p
obtained
by
taking
the
image
of
the
intersection
in
H
1
(G
k
i
,
T
i
)
with
the
image
of
H
1
(G
k
i
,
T
i
)
of
the
closure
[cf.
our
Galois-density
assumption,
together
with
the
evident
p-adic
continuity
of
the
assignment
y
i
→
δ
y
i
,y
i
]
of
the
set
obtained
by
adding
g
Y
i
[where
g
Y
i
is
the
genus
of
Y
i
]
elements
of
the
form
M
i
·
δ
y
i
,y
i
[where
M
i
is
the
“M
”
of
Corollary
2.7,
(iii)]
yields
—
from
the
point
of
view
of
Corollary
2.7,
(iii)
—
a
subset
that
coincides
with
the
subset
“N
k
•
”
[when
“k
•
”
is
taken
to
be
k
i
]
constructed
in
Corollary
2.7,
(iii).
Thus,
by
allowing
the
“k
i
”
to
vary
over
arbitrary
corresponding
finite
extensions
⊆
k
i
of
k
i
,
we
obtain
uniform
toral
neighborhoods
of
the
G
i
that
are
compatible
with
α.
But
this
implies
that
α
G
is
uniformly
toral,
hence
completes
the
proof
of
Corollary
2.9.
Remark
2.9.1.
Corollary
2.9
may
be
regarded
as
a
generalization
of
the
[MLF
portion
of]
[Mzk14],
Corollary
2.2,
to
the
case
of
pro-Σ
[where
Σ
is
of
cardinality
≥
2
and
contains
the
residue
characteristic
—
that
is
to
say,
Σ
is
not
necessarily
the
set
of
all
primes]
geometric
fundamental
groups
of
not
necessarily
affine
hyperbolic
curves.
From
this
point
of
view,
it
is
interesting
to
note
that
in
the
theory
of
the
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
59
present
§2,
Lemma
2.6,
which,
as
is
discussed
in
Remark
2.6.2,
is
reminiscent
of
a
classical
argument
on
the
“nonexistence
of
Frobenius
liftings”,
takes
the
place
of
Lemma
4.7
of
[Tama1],
which
is
applied
in
[Mzk14],
Corollary
2.1,
to
reconstruct
the
additive
structure
of
the
fields
involved.
In
this
context,
we
observe
that
the
appearance
of
“Frobenius
endomorphisms”
in
Remark
2.6.2
is
interesting
in
light
of
the
discussion
of
“hidden
endomorphisms”
in
the
Introduction,
in
which
“Frobenius
endomorphisms”
also
appear.
Remark
2.9.2.
One
way
to
think
of
Corollary
2.9
is
as
the
statement
that:
The
“Section
Conjecture”
over
MLF’s
implies
the
“absolute
isomorphism
version
of
the
Grothendieck
Conjecture”
over
MLF’s.
Here,
we
recall
that
in
the
notation
of
Corollary
2.9,
the
“Section
Conjecture”
over
MLF’s
amounts
to
the
assertion
that
every
closed
subgroup
of
Π
X
i
that
maps
isomorphically
to
an
open
subgroup
of
Gal(k
i
/k
i
)
is
the
decomposition
group
asso-
ciated
to
a
closed
point
of
X
i
.
In
fact,
in
order
to
apply
Corollary
2.9,
a
“relatively
weak
version
of
the
Section
Conjecture”
is
sufficient
—
cf.
the
point
of
view
of
[Mzk8].
Remark
2.9.3.
The
issue
of
verifying
the
“point-theoreticity
hypothesis”
of
Corol-
lary
2.9
[i.e.,
the
hypothesis
concerning
the
preservation
of
decomposition
groups
of
closed
points]
may
be
thought
of
as
consisting
of
two
steps,
as
follows:
(a)
First,
one
must
show
the
J-geometricity
[cf.
[Mzk3],
Definition
4.3]
of
the
image
via
α
of
a
decomposition
group
D
ξ
⊆
Π
X
1
of
a
closed
point
ξ
∈
X
1
(k
1
).
Once
one
shows
this
J-geometricity
for
all
finite
étale
coverings
of
X
2
arising
from
open
subgroups
of
Π
X
2
,
one
concludes
[cf.
the
arguments
of
[Mzk3],
§7,
§8]
that
there
exist
rational
points
of
a
certain
tower
of
coverings
of
X
2
determined
by
α(D
ξ
)
⊆
Π
X
2
over
tame
extensions
of
k
2
.
(b)
Finally,
one
must
show
that
these
rational
points
over
tame
extensions
of
k
2
necessarily
converge
—
an
issue
that
the
author
typically
refers
to
by
the
term
“tame
convergence”.
At
the
time
of
writing,
it
is
not
clear
to
the
author
how
to
complete
either
of
these
two
steps.
On
the
other
hand,
in
the
“birational”
—
or,
more
generally,
the
“co-Galois-dense”
—
case,
one
has
Corollary
2.10
[given
below].
Remark
2.9.4.
(i)
By
contrast
to
the
quite
substantial
difficulty
[discussed
in
Remark
2.9.3]
of
verifying
“point-theoreticity”
for
hyperbolic
curves
over
MLF’s,
in
the
case
of
hyperbolic
curves
over
finite
fields,
there
is
a
[relatively
simple]
“group-theoretic”
algorithm
for
reconstructing
the
decomposition
groups
of
closed
points,
which
follows
essentially
from
the
theory
of
[Tama1]
[cf.
[Tama1],
Corollary
2.10,
Proposition
3.8].
Such
an
algorithm
is
discussed
in
[Mzk14],
Remark
10,
although
the
argument
given
there
is
somewhat
sketchy
and
a
bit
misleading.
A
more
detailed
presentation
may
be
found
in
[SdTm],
Corollary
1.25.
60
SHINICHI
MOCHIZUKI
(ii)
A
more
concise
version
of
this
argument,
along
the
lines
of
[Mzk14],
Remark
10,
may
be
given
as
follows:
Let
X
be
a
proper
[for
simplicity]
hyperbolic
curve
over
a
finite
field
k,
with
algebraic
closure
k;
Σ
a
set
of
prime
numbers
that
contains
a
prime
that
is
invertible
in
k;
π
1
(X
×
k
k)
Δ
X
the
maximal
pro-Σ
quotient
of
the
étale
fundamental
group
π
1
(X
×
k
k)
of
X
×
k
k;
π
1
(X)
Π
X
the
corresponding
def
quotient
of
the
étale
fundamental
group
π
1
(X)
of
X;
Π
X
G
k
=
Gal(k/k)
the
natural
quotient.
Then
it
suffices
to
give
a
“group-theoretic”
characterization
of
the
quasi-sections
D
⊆
Π
X
[i.e.,
closed
subgroups
that
map
isomorphically
onto
an
open
subgroup
of
G
k
]
which
are
decomposition
groups
of
closed
points
of
X.
Write
→
X
X
for
the
pro-finite
étale
covering
corresponding
to
Π
X
.
If
E
⊆
Π
X
is
a
closed
subgroup
whose
image
in
G
k
is
open,
then
let
us
write
k
E
for
the
finite
extension
field
of
k
determined
by
this
image.
If
J
⊆
Π
X
is
an
open
subgroup,
then
let
us
write
def
X
J
→
X
for
the
covering
determined
by
J
and
J
Δ
=
J
Δ
X
.
If
J
⊆
Π
X
is
an
open
subgroup
such
that
J
Δ
is
a
characteristic
subgroup
of
Δ
X
,
then
we
shall
say
that
J
is
geometrically
characteristic.
Now
let
J
⊆
Π
X
be
a
geometrically
characteristic
open
subgroup.
Let
us
refer
to
as
a
descent-group
for
J
any
open
subgroup
H
⊆
Π
X
such
that
J
⊆
H,
J
Δ
=
H
Δ
.
Thus,
a
descent-group
H
for
J
may
be
thought
of
as
an
intermediate
covering
X
J
→
X
H
→
X
such
that
X
H
×
k
H
k
J
∼
=
X
J
.
Write
X
J
(k
J
)
fld-def
⊆
X
J
(k
J
)
for
the
subset
of
k
J
-valued
points
of
X
J
that
do
not
arise
from
points
∈
X
H
(k
H
)
for
any
descent-group
H
=
J
for
J
—
i.e.,
the
k
J
-valued
points
whose
field
of
definition
is
k
J
with
respect
to
all
possible
“descended
forms”
of
X
J
.
[That
is
to
say,
this
definition
of
“fld-def”
differs
slightly
from
the
definition
of
“fld-def”
in
[Mzk14],
that
maps
to
x
∈
X
J
(k
J
),
and
we
Remark
10.]
Thus,
if
x
is
a
closed
point
of
X
⊆
Π
X
for
the
stabilizer
in
Π
X
[i.e.,
“decomposition
group”]
of
x
,
then
it
write
D
x
def
·
J
Δ
(⊇
J)
[so
H
x
is
a
tautology
that
x
maps
to
a
point
∈
X
H
x
(k
H
x
)
for
H
x
=
D
x
forms
a
descent-group
for
J];
in
particular,
it
follows
immediately
that:
x
∈
X
J
(k
J
)
fld-def
⇐⇒
D
⊆
J
x
⇐⇒
H
x
=
J.
Now
it
follows
immediately
from
this
characterization
of
“fld-def”
that
if
J
1
⊆
J
2
⊆
Π
X
are
geometrically
characteristic
open
subgroups
such
that
k
J
1
=
k
J
2
,
then
the
natural
map
X
J
1
(k
J
1
)
→
X
J
2
(k
J
2
)
induces
a
map
X
J
1
(k
J
1
)
fld-def
→
X
J
2
(k
J
2
)
fld-def
.
Moreover,
these
considerations
allow
one
to
conclude
[cf.
the
theory
of
[Tama1]]
that:
A
quasi-section
D
⊆
Π
X
is
a
decomposition
group
of
a
closed
point
of
X
if
and
and
only
if,
for
every
geometrically
characteristic
open
subgroup
J
⊆
Π
X
such
that
D
·
J
Δ
=
J,
it
holds
that
X
J
(k
J
)
fld-def
=
∅.
Thus,
to
render
this
characterization
of
decomposition
groups
“group-theoretic”,
it
suffices
to
give
a
“group-theoretic”
criterion
for
the
condition
that
X
J
(k
J
)
fld-def
=
∅.
In
[Tama1],
the
Lefschetz
trace
formula
is
applied
to
compute
the
cardinality
of
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
61
X
J
(k
J
).
On
the
other
hand,
if
we
use
the
notation
“|
−
|”
to
denote
the
cardinality
of
a
finite
set,
then
one
verifies
immediately
that
|X
H
(k
H
)
fld-def
|
|X
J
(k
J
)|
=
H
—
where
H
⊇
J
ranges
over
the
descent-groups
for
J.
In
particular,
by
applying
induction
on
[Π
X
:
J],
one
concludes
immediately
from
the
above
formula
that
|X
J
(k
J
)
fld-def
|
may
be
computed
from
the
|X
H
(k
H
)|,
as
H
ranges
over
the
descent-
groups
for
J
[while
|X
H
(k
H
)|
may
be
computed,
as
in
[Tama1],
from
the
Lefschetz
trace
formula].
This
yields
the
desired
“group-theoretic”
characterization
of
the
decomposition
groups
of
Π
X
.
Corollary
2.10.
(Geometricity
of
Absolute
Isomorphisms
for
Co-
Galois-dense
Pro-curves)
For
i
=
1,
2,
let
k
i
be
an
MLF
of
residue
charac-
teristic
p
i
;
k
i
an
algebraic
closure
of
k
i
;
Σ
i
a
set
of
primes
of
cardinality
≥
2
such
that
p
i
∈
Σ
i
;
U
i
a
co-Galois-dense
pro-curve
over
k
i
.
Write
“π
1
(−)”
for
the
étale
fundamental
group
of
a
connected
scheme,
relative
to
an
appropriate
choice
of
basepoint;
Δ
U
i
for
the
maximal
pro-Σ
i
quotient
of
π
1
(U
i
×
k
i
k
i
);
Π
U
i
for
the
quotient
of
π
1
(U
i
)
by
the
kernel
of
the
natural
surjection
π
1
(U
i
×
k
i
k
i
)
Δ
U
i
.
Let
∼
α
:
Π
U
1
→
Π
U
2
be
an
isomorphism
of
profinite
groups.
Then
Δ
U
i
,
Π
U
i
are
slim;
p
1
=
p
2
;
Σ
1
=
Σ
2
;
α
is
geometric,
i.e.,
arises
from
a
unique
isomorphism
of
schemes
∼
U
1
→
U
2
.
Proof.
First,
we
observe
that
Σ
i
may
be
characterized
as
the
set
of
primes
l
such
that
Π
U
i
has
l-cohomological
dimension
>
2.
Thus,
Σ
1
=
Σ
2
.
Let
us
write
def
Σ
=
Σ
1
=
Σ
2
;
X
i
for
the
smooth,
proper,
geometrically
connected
curve
over
k
i
determined
by
U
i
;
Δ
X
i
for
the
maximal
pro-Σ
quotient
of
π
1
(X
i
×
k
i
k
i
);
Π
X
i
for
the
quotient
of
π
1
(X
i
)
by
Ker(π
1
(X
i
×
k
i
k
i
)
Δ
X
i
).
Thus,
U
i
determines
some
Galois-dense
subset
Ξ
i
⊆
X
i
(k
i
).
Since
Δ
U
i
,
Π
U
i
may
be
written
as
inverse
limits
of
surjections
of
slim
profinite
groups
[cf.,
e.g.,
[Mzk15],
Proposition
2.3],
it
follows
that
Δ
U
i
,
Π
U
i
are
slim.
Since
the
kernel
of
the
natural
surjection
Π
U
i
Π
X
i
is
topologically
generated
by
the
inertia
groups
of
points
∈
Ξ
i
,
and
these
inertia
Σ
(1)
[where
the
“(1)”
denotes
a
Tate
twist;
we
write
groups
are
isomorphic
to
Z
Σ
for
the
maximal
pro-Σ
quotient
of
Z]
in
a
fashion
that
is
compatible
with
the
Z
def
conjugation
action
of
some
open
subgroup
of
G
i
=
Gal(k
i
/k
i
),
it
follows
that
we
obtain
an
isomorphism
∼
ab-t
Π
ab-t
U
i
→
Π
X
i
on
torsion-free
abelianizations.
In
particular,
it
follows
[in
light
of
our
assumptions
on
Σ
i
]
that,
in
the
notation
of
[Mzk15],
Theorem
2.6,
sup
p
,p
∈Σ
{δ
p
1
(Π
U
i
)
−
δ
p
1
(Π
U
i
)}
=
sup
p
,p
∈Σ
=
sup
p
,p
∈Σ
{δ
p
1
(Π
X
i
)
−
δ
p
1
(Π
X
i
)}
{δ
p
1
(G
i
)
−
δ
p
1
(G
i
)}
=
[k
i
:
Q
p
i
]
62
SHINICHI
MOCHIZUKI
—
cf.
[Mzk15],
Theorem
2.6,
(ii).
In
particular,
by
applying
this
chain
of
equalities
to
arbitrary
open
subgroups
of
Π
U
i
,
we
conclude
that
α
induces
an
isomorphism
∼
α
G
:
G
1
→
G
2
.
Moreover,
by
[Mzk5],
Proposition
1.2.1,
(i),
(vi),
the
existence
of
def
α
G
implies
that
p
1
=
p
2
[so
we
set
p
=
p
1
=
p
2
],
and
α
G
is
compatible
with
the
respective
cyclotomic
characters.
Let
M
be
a
profinite
abelian
group
equipped
with
a
continuous
H-action,
for
H
⊆
G
i
[where
i
∈
{1,
2}]
an
open
subgroup.
Then
let
us
write
M
Q
(M
)
for
the
quotient
of
M
by
the
closed
subgroup
generated
by
the
quasi-toral
subgroups
of
M
[i.e.,
closed
subgroups
isomorphic
as
J-modules,
for
J
⊆
H
an
open
subgroup,
to
Z
l
(1)
for
some
prime
l];
M
Q
(M
)
Q(M
)
for
the
maximal
torsion-free
quotient
of
Q
(M
).
Also,
if
M
is
topologically
finitely
generated,
then
let
us
write
M
T
(M
)
for
the
maximal
torsion-free
quasi-trivial
quotient
[i.e.,
maximal
torsion-free
quotient
on
which
H
acts
through
a
finite
quotient].
Then
one
verifies
immediately
that
the
assignments
M
→
Q(M
),
M
→
T
(M
)
are
functorial.
Moreover,
it
follows
from
the
observations
of
the
preceding
paragraph
that
the
natural
surjection
Δ
U
i
Δ
ab-t
Δ
X
i
determines
a
surjection
on
torsion-free
abelianizations
Δ
ab-t
U
i
X
i
that
∼
ab-t
ab-t
induces
an
isomorphism
Q(Δ
U
i
)
→
Q(Δ
X
i
).
Thus,
it
follows
from
“Poincaré
∼
ab-t
duality”
[i.e.,
the
isomorphism
Δ
ab-t
X
i
→
Hom(Δ
X
i
,
Z(1))
determined
by
the
cup-
product
on
the
étale
cohomology
of
X]
that
ab-t
2g
X
i
=
dim
Q
l
(Q(Δ
ab-t
X
i
)
⊗
Q
l
)
+
dim
Q
l
(T
(Δ
X
i
)
⊗
Q
l
)
ab-t
=
dim
Q
l
(Q(Δ
ab-t
X
i
)
⊗
Q
l
)
+
dim
Q
l
(T
(Q(Δ
X
i
))
⊗
Q
l
)
ab-t
=
dim
Q
l
(Q(Δ
ab-t
U
i
)
⊗
Q
l
)
+
dim
Q
l
(T
(Q(Δ
U
i
))
⊗
Q
l
)
—
where
g
X
i
is
the
genus
of
X
i
,
and
l
∈
Σ.
Thus,
we
conclude
that
g
X
1
=
g
X
2
.
In
particular,
by
passing
to
corresponding
[i.e.,
via
α]
open
subgroups
of
the
Π
U
i
,
we
may
assume
that
g
X
1
=
g
X
2
≥
2.
Next,
by
applying
this
equality
“g
X
1
=
g
X
2
”
to
corresponding
[i.e.,
via
α]
open
subgroups
of
the
Π
U
i
,
it
follows
from
the
Hurwitz
formula
that
the
condition
on
a
pair
of
open
subgroups
J
i
⊆
H
i
⊆
Δ
U
i
that
“the
covering
between
J
i
and
H
i
be
cyclic
of
order
a
power
of
a
prime
number
and
totally
ramified
at
precisely
one
closed
point
but
unramified
elsewhere”
is
preserved
by
α.
Thus,
it
follows
formally
[cf.,
e.g.,
the
latter
portion
of
the
proof
of
[Mzk5],
Lemma
1.3.9]
that
α
preserves
the
inertia
groups
of
points
∈
Ξ
i
.
Moreover,
by
considering
the
conjugation
action
of
Π
U
i
on
these
inertia
groups,
we
conclude
that
α
preserves
the
decomposition
groups
∼
⊆
Π
U
i
of
points
∈
Ξ
i
.
Thus,
in
summary,
α
induces
an
isomorphism
Π
X
1
→
Π
X
2
that
preserves
the
decomposition
groups
⊆
Π
X
i
of
points
∈
Ξ
i
;
in
particular,
by
∼
applying
Corollary
2.9
to
this
isomorphism
Π
X
1
→
Π
X
2
,
we
obtain
an
isomorphism
∼
of
schemes
U
1
→
U
2
,
as
desired.
This
completes
the
proof
of
Corollary
2.10.
Remark
2.10.1.
Thus,
by
contrast
to
the
results
of
[Mzk14],
Corollary
2.3,
or
[Mzk15],
Corollary
3.8,
(g)
[cf.
[Mzk15],
Remark
3.8.1]
—
or,
indeed,
Corollary
1.11,
(iii)
of
the
present
paper
—
Corollary
2.10
constitutes
the
first
“absolute
isomorphism
version
of
the
Grothendieck
Conjecture
over
MLF’s”
known
to
the
author
that
does
not
rely
on
the
use
of
Belyi
maps.
One
aspect
of
this
independence
of
the
theory
of
Belyi
maps
may
be
seen
in
the
fact
that
Corollary
2.10
may
be
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
63
applied
to
pro-curves
which
are
not
of
unit
type
[cf.
Remark
2.8.2,
(i),
(iii)].
Another
aspect
of
this
independence
of
the
theory
of
Belyi
maps
may
be
seen
in
the
fact
that
Corollary
2.10
involves
geometrically
pro-Σ
arithmetic
fundamental
groups
for
Σ
which
are
not
necessarily
equal
to
the
set
of
all
prime
numbers.
Finally,
we
observe
that
the
techniques
developed
in
the
present
§2
allow
one
to
give
a
more
pedestrian
treatment
of
the
[somewhat
sketchy]
treatment
given
in
[Mzk9]
[cf.
the
verification
of
“observation
(iv)”
given
in
the
proof
of
[Mzk9],
Corollary
3.11,
as
well
as
Remark
2.11.1
below]
of
the
fact
that
“cusps
always
appear
as
images
of
nodes”.
Corollary
2.11.
(Cusps
as
Images
of
Nodes)
Let
k
be
a
complete
dis-
cretely
valued
field
of
characteristic
zero,
with
perfect
residue
field
k
of
char-
def
acteristic
p
>
0
and
ring
of
integers
O
K
;
η
=
Spec(k);
S
log
the
log
scheme
obtained
def
by
equipping
S
=
Spec(O
K
)
with
the
log
structure
determined
by
the
closed
point
def
S
=
Spec(k)
of
S;
X
log
→
S
log
a
stable
log
curve
over
S
log
such
that
the
un-
derlying
scheme
of
the
generic
fiber
X
log
×
S
η
is
smooth;
ξ
∈
X(S)
a
cusp
of
the
stable
log
curve
X
log
;
ξ
∈
X(S)
the
restriction
of
ξ
to
the
special
fiber
X
of
X.
In
the
following,
we
shall
denote
restrictions
to
η
by
means
of
a
subscript
η;
also
we
shall
often
identify
ξ
with
its
image
in
X.
Then,
after
possibly
replacing
k
by
a
finite
extension
of
k,
there
exists
a
morphism
of
stable
log
curves
over
S
log
φ
log
:
Y
log
→
X
log
such
that
the
following
properties
are
satisfied:
log
(a)
the
restriction
φ
log
→
X
η
log
is
a
finite
log
étale
Galois
covering;
η
:
Y
η
(b)
ξ
is
the
image
of
a
node
of
the
special
fiber
Y
of
Y
;
(c)
ξ
is
the
image
of
an
irreducible
component
of
Y
.
If,
moreover,
X
is
sturdy,
loop-ample,
and
singular,
then
φ
η
:
Y
η
→
X
η
may
be
taken
to
be
finite
étale
of
degree
p.
Proof.
By
replacing
k
by
an
appropriate
subfield
of
k,
one
verifies
immediately
that
we
may
assume
that
k
is
of
countable
cardinality,
hence
that
k
satisfies
the
hypotheses
of
the
discussion
preceding
Lemma
2.6.
After
possibly
replacing
k
by
a
finite
extension
of
k
and
X
η
log
by
a
finite
log
étale
Galois
covering
of
X
η
log
[which,
in
fact,
may
be
taken
to
be
of
degree
a
power
of
p
·
l,
where
l
is
a
prime
=
p],
we
may
assume
that
X
is
sturdy
[cf.
§0],
loop-ample
[cf.
§0],
singular
[cf.
Remark
2.6.3],
and
split.
Next,
let
us
recall
from
the
well-known
theory
of
pointed
stable
curves
[cf.
[Knud]]
that
if
we
write
V
log
→
S
log
for
the
stable
log
curve
obtained
by
forgetting
the
cusps
of
X
log
[so
V
η
=
X
η
],
then
it
follows
immediately
from
the
fact
that
X
is
sturdy
that
V
=
X.
Thus,
by
Lemma
2.6,
(i)
[cf.
also
the
way
in
which
Lemma
2.6,
(i),
is
applied
in
the
proof
of
Lemma
2.6,
(ii)],
it
follows
that,
after
64
SHINICHI
MOCHIZUKI
possibly
replacing
k
by
a
finite
extension
of
k,
there
exists
a
morphism
of
stable
log
curves
over
S
log
φ
log
:
Y
log
→
X
log
such
that
if
we
write
φ
for
the
morphism
of
schemes
underlying
φ
log
,
then
φ
η
:
Y
η
→
X
η
is
a
finite
étale
Galois
covering
of
degree
p
that
is
wildly
ramified
over
the
irreducible
component
C
of
X
containing
ξ.
Next,
let
us
suppose
that
the
property
(c)
is
satisfied.
Thus,
there
exists
an
irreducible
component
E
of
Y
that
maps
to
ξ.
Next,
let
us
observe
that
there
exists
an
irreducible
component
D
of
Y
that
maps
finitely
to
C
and
meets
the
connected
component
F
of
the
fiber
φ
−1
(ξ)
that
contains
E
[so
D
is
not
contained
in
F
].
In
particular,
it
follows
that
there
exists
a
chain
of
irreducible
components
E
1
=
E,
E
2
,
.
.
.
,
E
n
[where
n
≥
1
is
an
integer]
of
Y
joining
E
to
D
such
that
each
E
j
⊆
F
[for
j
=
1,
.
.
.
,
n].
Thus,
E
n
meets
D
at
some
node
of
Y
that
maps
to
ξ.
That
is
to
say,
property
(b)
is
satisfied.
Thus,
to
complete
the
proof
of
Corollary
2.11,
it
suffices
to
verify
property
(c).
Now
suppose
that
property
(c)
fails
to
hold.
Then
φ
is
finite
over
some
neigh-
borhood
of
ξ.
Since
φ
η
is
wildly
ramified
over
C,
it
follows
that
there
exists
a
nontrivial
element
σ
∈
Gal(Y
η
log
/X
η
log
)
that
fixes
and
acts
as
the
identity
on
some
irreducible
component
D
of
Y
that
maps
finitely
to
C.
After
possibly
replacing
k
by
a
finite
extension
of
k,
it
follows
from
the
finiteness
of
φ
over
some
neighborhood
of
ξ
that
we
may
assume
that
there
exists
a
cusp
ζ
∈
Y
(S)
of
Y
log
lying
over
ξ
such
that
the
restriction
ζ
of
ζ
to
Y
lies
in
D.
But
then
the
distinct
[since
σ
is
nontrivial,
and
φ
η
is
étale]
cusps
ζ,
ζ
σ
of
Y
log
have
identical
restrictions
ζ,
ζ
σ
to
Y
—
in
contradiction
to
the
definition
of
a
“stable
log
curve”
[i.e.,
of
a
“pointed
stable
curve”].
This
completes
the
proof
of
property
(c)
and
hence
of
Corollary
2.11.
Remark
2.11.1.
(i)
The
statement
of
[Mzk9],
Corollary
3.11,
concerns
smooth
log
curves
over
an
MLF,
but
in
fact,
the
same
proof
as
the
proof
given
in
[Mzk9]
for
[Mzk9],
Corollary
3.11,
may
be
applied
to
smooth
log
curves
over
an
arbitrary
mixed
characteristic
complete
discretely
valued
field.
Here,
we
note
that
by
passing
to
an
appropriate
extension,
this
discretely
valued
field
may
be
assumed
to
have
a
perfect
residue
field,
as
in
Corollary
2.11.
In
particular,
Corollary
2.11
may
be
applied
to
[the
portion
corresponding
to
“observation
(iv)”
in
loc.
cit.
of]
the
proof
of
such
a
generalization
of
[Mzk9],
Corollary
3.11,
for
more
general
fields.
(ii)
In
the
discussion
of
the
“pro-Σ
version”
of
[Mzk9],
Corollary
3.11,
in
[Mzk9],
Remark
3.11.1,
one
should
assume
that
p
α
,
p
β
∈
Σ.
In
fact,
this
assumption
is,
in
some
sense,
implicit
in
the
phraseology
that
appears
in
the
first
two
lines
of
[Mzk9],
Remark
3.11.1,
but
it
should
have
been
stated
explicitly.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
65
Section
3:
Elliptic
and
Belyi
Cuspidalizations
The
sort
of
preservation
of
decomposition
groups
of
closed
points
that
is
re-
quired
in
the
hypothesis
of
Corollary
2.9
is
shown
[for
certain
types
of
hyperbolic
curves]
in
the
case
of
profinite
geometric
fundamental
groups
in
[Mzk8],
Corollary
3.2.
On
the
other
hand,
at
the
time
of
writing,
the
author
does
not
know
of
any
such
results
in
the
case
of
pro-Σ
geometric
fundamental
groups,
when
Σ
is
not
equal
to
the
set
of
all
primes.
Nevertheless,
in
the
present
§3,
we
observe
that
the
tech-
niques
of
[Mzk8],
§2,
concerning
the
preservation
of
decomposition
groups
of
torsion
points
of
elliptic
curves
do
indeed
hold
for
fairly
general
pro-Σ
geometric
fundamen-
tal
groups
[cf.
Corollaries
3.3,
3.4].
Moreover,
we
observe
that
these
techniques
—
which
may
be
applied
not
only
to
[hyperbolic
orbicurves
related
to]
elliptic
curves,
but
also,
in
the
profinite
case,
to
[hyperbolic
orbicurves
related
to]
tripods
[i.e.,
hyperbolic
curves
of
type
(0,
3)
—
cf.
[Mzk15],
§0],
via
the
use
of
Belyi
maps
—
allow
one
to
recover
not
only
the
decomposition
groups
of
[certain]
closed
points,
but
also
the
resulting
“cuspidalizations”
[i.e.,
the
arithmetic
fundamental
groups
of
open
subschemes
obtained
by
removing
such
closed
points]
—
cf.
Corollaries
3.7,
3.8.
Let
X
be
a
hyperbolic
orbicurve
over
a
field
k
of
characteristic
zero;
k
an
algebraic
closure
of
k.
We
shall
denote
the
base-change
operation
“×
k
k”
by
means
of
a
subscript
k.
Thus,
we
have
an
exact
sequence
of
fundamental
groups
1
→
π
1
(X
k
)
→
π
1
(X)
→
Gal(k/k)
→
1.
Definition
3.1.
Let
π
1
(X)
Π
be
a
quotient
of
profinite
groups.
Write
Δ
⊆
Π
for
the
image
of
π
1
(X
k
)
in
Π.
Then
we
shall
say
that
X
is
Π-elliptically
admissible
if
the
following
conditions
hold:
(a)
X
admits
a
k-core
[in
the
sense
of
[Mzk6],
Remark
2.1.1]
X
→
C;
(b)
C
is
semi-elliptic
[cf.
§0],
hence
admits
a
double
covering
D
→
C
by
a
once-punctured
elliptic
curve
D;
(c)
X
admits
a
finite
étale
covering
Y
→
X
by
a
hyperbolic
curve
Y
over
a
finite
extension
k
Y
of
k
that
arises
from
a
normal
open
subgroup
Π
Y
⊆
Π
such
that
the
resulting
finite
étale
covering
Y
→
C
factors
as
the
composite
of
a
covering
Y
→
D
with
the
covering
D
→
C
and,
moreover,
is
such
that,
for
every
set
of
primes
Σ
such
that
some
open
subgroup
of
Δ
is
pro-Σ,
it
holds
that
Δ
is
pro-Σ,
and,
moreover,
the
degree
of
the
covering
Y
→
C
×
k
k
Y
is
a
product
of
primes
[perhaps
with
multiplicities]
∈
Σ.
When
Π
=
π
1
(X),
we
shall
simply
say
that
X
is
elliptically
admissible.
Remark
3.1.1.
In
the
notation
of
Definition
3.1,
one
verifies
immediately
that
D
k
→
C
k
may
be
characterized
as
the
unique
[up
to
isomorphism
over
C
k
]
finite
étale
double
covering
of
C
k
by
a
hyperbolic
curve
[i.e.,
as
opposed
to
an
arbitrary
hyperbolic
orbicurve].
66
SHINICHI
MOCHIZUKI
Example
3.2.
Scheme-theoretic
Elliptic
Cuspidalizations.
(i)
Let
N
be
a
positive
integer;
D
a
once-punctured
elliptic
curve
over
a
finite
Galois
extension
k
of
k
such
that
all
of
the
N
-torsion
points
of
the
underlying
elliptic
curve
E
of
D
are
defined
over
k
;
D
→
C
a
semi-elliptic
k
-core
of
D
[such
that
D
→
C
is
the
double
covering
appearing
in
the
definition
of
“semi-elliptic”].
Then
the
morphism
[N
]
E
:
E
→
E
given
by
multiplication
by
N
determines
a
finite
étale
covering
[N
]
D
:
U
→
D
[of
degree
N
2
],
together
with
an
open
embedding
U
→
D
[which
we
use
to
identify
U
with
its
image
in
D],
i.e.,
we
have
a
diagram
as
follows:
U
→
D
⏐
⏐
[N
]
D
D
Suppose
that
the
Galois
group
Gal(k/k)
is
slim.
Then,
in
the
language
of
[Mzk15],
§4,
this
situation
may
be
described
as
follows
[cf.
[Mzk15],
Definition
4.2,
(i),
where
we
take
the
extension
“1
→
Δ
→
Π
→
G
→
1”
to
be
the
extension
1
→
π
1
(D
×
k
k)
→
π
1
(D)
→
Gal(k/k
)
→
1]:
The
above
diagram
yields
a
chain
D
U
(→
D)
(U
→)
U
n
(U
n
→)
U
n−1
.
.
.
def
(U
3
→)
U
2
(U
2
→)
U
1
=
D
def
[where
n
=
N
2
−
1]
whose
associated
type-chain
is
,
•,
.
.
.
,
•
[i.e.,
a
finite
étale
covering,
followed
by
n
de-cuspidalizations],
together
with
a
terminal
isomorphism
∼
U
1
→
D
[which,
in
our
notation,
amounts
to
the
identity
morphism]
from
the
U
1
at
the
end
of
the
above
chain
to
the
unique
D
of
the
trivial
chain
[of
length
0].
In
particular:
The
above
chain
may
thought
of
as
a
construction
of
a
“cuspidalization”
[i.e.,
result
of
passing
to
an
open
subscheme
by
removing
various
closed
points]
U
→
D
of
D.
The
remainder
of
the
portion
of
the
theory
of
the
present
§3
concerning
elliptic
cuspidalizations
consists,
in
essence,
of
the
unraveling
of
various
consequences
of
this
“chain-theoretic
formulation”
of
the
diagram
that
appears
at
the
beginning
of
the
present
item
(i).
(ii)
A
variant
of
the
discussion
of
(i)
may
be
obtained
as
follows.
In
the
notation
of
(i),
suppose
further
that
X
is
an
elliptically
admissible
hyperbolic
orbicurve
over
k,
and
that
we
have
been
given
finite
étale
coverings
V
→
X,
V
→
D,
where
V
is
a
hyperbolic
curve
over
k
.
Also,
[for
simplicity]
we
suppose
that
V
→
X
is
a
Galois
def
covering
such
that
Gal(V
/X)
preserves
the
open
subscheme
U
V
=
V
×
D
U
⊆
V
[i.e.,
the
inverse
image
of
U
⊆
D
via
V
→
D].
Thus,
U
V
⊆
V
descends
to
an
open
subscheme
U
X
⊆
X.
Then
by
appending
to
the
chain
of
(i)
the
“finite
étale
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
67
covering”
V
→
X,
followed
by
the
“finite
étale
quotient”
V
→
D
on
the
left,
and
the
“finite
étale
covering”
V
→
D,
followed
by
the
“finite
étale
quotient”
V
→
X
on
the
right,
we
obtain
a
chain
X
V
(→
X)
(V
→)
D
U
(→
D)
(U
→)
U
n
(U
n
→)
U
n−1
.
.
.
def
def
(U
3
→)
U
2
(U
2
→)
U
1
=
D
V
(→
D)
(V
→)
X
∗
=
X
whose
associated
type-chain
is
,
,
,
•,
.
.
.
,
•,
,
∼
[where
the
“.
.
.
”
are
all
“•’s”],
together
with
a
terminal
isomorphism
X
∗
→
X
[i.e.,
the
identity
morphism].
In
particular,
the
above
chain
may
thought
of
as
a
construction
of
a
“cuspidalization”
U
X
→
X
of
X
via
the
construction
of
a
“cuspi-
dalization”
U
V
→
V
of
V
,
equipped
with
descent
data
[i.e.,
a
suitable
collection
of
automorphisms]
with
respect
to
the
finite
étale
Galois
covering
V
→
X.
Now
by
translating
the
scheme-theoretic
discussion
of
Example
3.2
into
the
language
of
profinite
groups
via
the
theory
of
[Mzk15],
§4,
we
obtain
the
following
result.
Corollary
3.3.
(Pro-Σ
Elliptic
Cuspidalization
I:
Algorithms)
Let
D
be
a
chain-full
set
of
collections
of
partial
construction
data
[cf.
[Mzk15],
Definition
4.6,
(i)]
such
that
the
rel-isom-DGC
holds
[i.e.,
the
“relative
isomorphism
version
of
the
Grothendieck
Conjecture
for
D
holds”
—
cf.
[Mzk15],
Definition
4.6,
(ii)];
G
a
slim
profinite
group;
1
→
Δ
→
Π
→
G
→
1
an
extension
of
GSAFG-type
that
admits
partial
construction
data
(k,
X,
Σ),
where
k
is
of
characteristic
zero,
and
X
is
a
Π-elliptically
admissible
[cf.
Def-
inition
3.1]
hyperbolic
orbicurve,
such
that
([X],
[k],
Σ)
∈
D;
α
:
π
1
(X)
Π
the
corresponding
scheme-theoretic
envelope
[cf.
[Mzk15],
Definition
2.1,
(iii)];
∼
→
X
the
pro-finite
étale
covering
of
X
determined
by
α
[so
Π
→
Gal(
X/X)];
X
∼
k
the
resulting
field
extension
of
k
[so
G
→
Gal(
k/k)].
Suppose
further
that,
for
×
some
l
∈
Σ,
the
cyclotomic
character
G
→
Z
l
has
open
image.
Thus,
by
the
theory
of
[Mzk15],
§4,
we
have
associated
categories
Chain(Π);
Chain
iso-trm
(Π);
ÉtLoc(Π)
which
may
be
constructed
via
purely
“group-theoretic”
operations
from
the
extension
of
profinite
groups
1
→
Δ
→
Π
→
G
→
1
[cf.
[Mzk15],
Definition
4.2,
(iii),
(iv),
(v);
[Mzk15],
Lemma
4.5,
(v);
the
proof
of
[Mzk15],
Theorem
4.7,
(ii)].
Then:
(i)
Let
G
⊆
G
be
a
normal
open
subgroup,
corresponding
to
some
finite
ex-
def
def
k
of
k;
Π
=
Π
×
G
G
;
C
a
k
-core
of
X
k
=
X
×
k
k
.
Then
tension
k
⊆
the
finite
étale
covering
X
k
→
C
determines
a
chain
X
k
C
of
the
category
k
)
[cf.
[Mzk15],
Definition
4.2,
(i),
(ii)]
whose
image
Π
Π
C
in
Chain(
X/X
Chain(Π
)
[via
the
natural
functor
of
[Mzk15],
Remark
4.2.1]
may
be
characterized
68
SHINICHI
MOCHIZUKI
“group-theoretically”,
up
to
isomorphism
in
Chain(Π
),
as
the
unique
chain
of
length
1
in
Chain(Π
),
with
associated
type-chain
,
such
that
the
resulting
object
of
ÉtLoc(Π
)
forms
a
terminal
object
of
ÉtLoc(Π
).
(ii)
The
collection
of
open
subgroups
Π
D
⊆
Π
C
that
arise
from
finite
étale
double
coverings
D
→
C
that
exhibit
C
as
semi-elliptic
[cf.
Remark
3.1.1]
may
be
characterized
“group-theoretically”
as
the
collection
of
open
subgroups
def
J
⊆
Π
C
of
index
2
such
that
J
Δ
C
[where
Δ
C
=
Ker(Π
C
G
)]
is
torsion-free
[i.e.,
the
covering
determined
by
J
is
a
scheme
—
cf.
[Mzk15],
Lemma
4.1,
(iv)].
def
(iii)
Write
Δ
D
=
Ker(Π
D
G
).
Let
N
be
a
positive
integer
which
is
a
product
of
primes
[perhaps
with
multipliticites]
of
Σ;
U
⊆
D
the
open
subscheme
obtained
by
removing
the
N
-torsion
points
of
the
elliptic
curve
underlying
D;
V
→
X,
V
→
D
finite
étale
coverings,
where
V
is
a
hyperbolic
curve
over
k
.
Suppose
further
that
V
→
X
arises
from
a
normal
open
subgroup
Π
V
⊆
Π
such
that
def
Gal(V
/X)
∼
=
Π/Π
V
preserves
the
open
subscheme
U
V
=
V
×
D
U
⊆
V
[i.e.,
the
inverse
image
of
U
⊆
D
via
V
→
D],
while
V
→
D
arises
from
an
open
immersion
Π
V
→
Π
D
.
Thus,
U
V
⊆
V
descends
to
an
open
subscheme
U
X
⊆
X,
and
U
⊆
D,
U
V
⊆
V
,
U
X
⊆
X
determine
extensions
of
GSAFG-type
1
→
Δ
U
→
Π
U
→
G
→
1;
1
→
Δ
U
V
→
Π
U
V
→
G
→
1
1
→
Δ
U
X
→
Π
U
X
→
G
→
1
[i.e.,
by
considering
the
finite
étale
Galois
coverings
of
degree
a
product
of
primes
[perhaps
with
multipliticites]
∈
Σ
over
coverings
of
U
,
U
V
,
U
X
arising
from
Π],
together
with
natural
surjections
Π
U
Π
D
,
Π
U
V
Π
V
,
Π
U
X
Π
and
open
im-
mersions
Π
U
V
→
Π
U
,
Π
U
V
→
Π
U
X
.
[In
particular,
Δ
U
,
Δ
U
V
,
and
Δ
U
X
are
pro-Σ
groups.]
Then,
for
any
G
⊆
G
that
is
sufficiently
small,
where
“sufficiently”
depends
only
on
N
,
the
natural
surjection
Π
U
X
Π
—
i.e.,
“cuspidalization”
of
Π
—
may
be
constructed
via
“group-theoretic”
operations
as
follows:
(a)
There
exists
a
[not
necessarily
unique]
Π-chain,
which
admits
an
en-
tirely
“group-theoretic”
description,
with
associated
type-chain
,
,
,
•,
.
.
.
,
•,
,
—
cf.
Example
3.2,
(ii)
—
that
admits
a
terminal
isomorphism
with
the
trivial
Π-chain
[of
length
0],
and
whose
final
three
groups
consist
of
Π
D
Π
V
(→
Π
D
)
(Π
V
→)
Π
such
that
the
natural
surjection
Π
U
Π
D
may
be
recovered
from
the
chain
of
“•’s”
terminating
at
the
third
to
last
group
of
the
above-mentioned
Π-chain;
the
natural
surjection
Π
U
V
Π
V
may
then
be
recovered
from
Π
U
Π
D
by
forming
the
fiber
product
with
the
inclusion
Π
V
→
Π
D
.
(b)
The
natural
surjection
Π
U
X
Π
may
be
recovered
from
Π
U
V
Π
V
[where
we
note
that
Π
U
V
Π
V
may
be
identified
with
the
fiber
product
of
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
69
out
Π
U
X
Π
with
the
inclusion
Π
V
→
Π]
by
forming
the
“
”
[cf.
§0]
with
respect
to
the
unique
lifting
[relative
to
Π
U
V
Π
V
]
of
the
outer
action
of
the
finite
group
Π/Π
V
on
Π
V
to
a
group
of
outer
automorphisms
of
Π
U
V
.
(c)
The
decomposition
groups
of
the
closed
points
of
X
lying
in
the
complement
of
U
X
may
be
obtained
as
the
images
via
Π
U
X
Π
of
the
cuspidal
decomposition
groups
of
Π
U
X
[cf.
[Mzk15],
Lemma
4.5,
(v)].
Proof.
The
assertions
of
Corollary
3.3
follow
immediately
from
the
definitions,
together
with
the
various
references
quoted
in
the
course
of
the
“group-theoretic”
reconstruction
algorithm
described
in
the
statement
of
Corollary
3.3,
and
the
equiv-
alences
of
[Mzk15],
Theorem
4.7,
(i).
Remark
3.3.1.
Let
p
be
a
prime
number.
Then
if
one
takes
F
to
be
set
of
iso-
morphism
classes
of
generalized
sub-p-adic
fields,
S
the
set
of
sets
of
prime
numbers
containing
p,
and
V
to
be
the
set
of
isomorphism
classes
of
hyperbolic
orbicurves
def
over
fields
whose
isomorphism
class
∈
F,
then
D
=
V×F×S
satisfies
the
hypothesis
of
Corollary
3.3
concerning
“D”
[cf.
[Mzk15],
Example
4.8,
(i)].
Remark
3.3.2.
Recall
that
when
k
is
an
MLF
or
an
NF,
the
subgroup
Δ
⊆
Π
admits
a
purely
“group-theoretic”
characterization
[cf.
[Mzk15],
Theorem
2.6,
(v),
(vi)].
Thus,
when
k
is
an
MLF
or
an
NF,
the
various
“group-theoretic”
reconstruc-
tion
algorithms
described
in
the
statement
of
Corollary
3.3
may
be
thought
of
as
being
applied
not
to
the
extension
1
→
Δ
→
Π
→
G
→
1,
but
rather
to
the
single
profinite
group
Π.
Remark
3.3.3.
One
verifies
immediately
that
Corollary
3.3
admits
a
“tempered
version”,
when
the
base
field
k
is
an
MLF
[cf.
[Mzk15],
Theorem
4.12,
(i)].
We
leave
the
routine
details
to
the
reader.
Remark
3.3.4.
By
applying
the
tempered
version
of
Corollary
3.3
discussed
in
Remark
3.3.3,
one
may
obtain
“explicit
reconstruction
algorithm
versions”
of
certain
results
of
[Mzk12]
[cf.
[Mzk12],
Theorem
1.6;
[Mzk12],
Remark
1.6.1]
concerning
the
étale
theta
function.
We
leave
the
routine
details
to
the
reader.
The
“group-theoretic”
algorithm
of
Corollary
3.3
has
the
following
immediate
“Grothendieck
Conjecture-style”
consequence.
Corollary
3.4.
(Pro-Σ
Elliptic
Cuspidalization
II:
Comparison)
Let
D
be
a
chain-full
set
of
collections
of
partial
construction
data
[cf.
[Mzk15],
Definition
4.6,
(i)]
such
that
the
rel-isom-DGC
holds
[cf.
[Mzk15],
Definition
4.6,
(ii)].
For
i
=
1,
2,
let
G
i
be
a
slim
profinite
group;
1
→
Δ
i
→
Π
i
→
G
i
→
1
70
SHINICHI
MOCHIZUKI
an
extension
of
GSAFG-type
that
admits
partial
construction
data
(k
i
,
X
i
,
Σ
i
),
where
k
i
is
of
characteristic
zero,
and
X
i
is
a
Π
i
-elliptically
admissible
[cf.
Definition
3.1]
hyperbolic
orbicurve,
such
that
([X
i
],
[k
i
],
Σ
i
)
∈
D;
α
i
:
π
1
(X
i
)
Π
i
the
corresponding
scheme-theoretic
envelope
[cf.
[Mzk15],
Defi-
i
→
X
i
the
pro-finite
étale
covering
of
X
determined
by
α
i
nition
2.1,
(iii)];
X
∼
∼
i
/X
i
)];
[so
Π
i
→
Gal(
X
k
i
the
resulting
field
extension
of
k
i
[so
G
i
→
Gal(
k
i
/k
i
)];
C
i
a
k
i
-core
of
X
i
;
D
i
→
C
i
a
finite
étale
double
covering
that
exhibits
C
i
as
semi-elliptic
[cf.
Remark
3.1.1];
Π
i
⊆
Π
C
i
,
Π
D
i
⊆
Π
C
i
the
open
subgroups
deter-
integer
which
is
a
product
of
primes
mined
by
X
i
→
C
i
,
D
i
→
C
i
;
N
a
positive
[perhaps
with
multipliticites]
∈
Σ
1
Σ
2
;
U
i
⊆
D
i
the
open
subscheme
obtained
by
removing
the
N
-torsion
points
of
the
elliptic
curve
underlying
D
i
;
V
i
→
X
i
,
V
i
→
D
i
finite
étale
coverings
that
arise
from
a
normal
open
subgroup
Π
V
i
⊆
Π
i
and
an
open
immersion
Π
V
i
→
Π
D
i
such
that
Gal(V
i
/X
i
)
∼
=
Π
i
/Π
V
i
preserves
def
the
open
subscheme
U
V
i
=
V
i
×
D
i
U
i
⊆
V
i
[i.e.,
the
inverse
image
of
U
i
⊆
D
i
via
V
i
→
D
i
];
U
X
i
⊆
X
i
the
resulting
open
subscheme
[obtained
by
descending
U
V
i
⊆
V
i
];
1
→
Δ
U
Xi
→
Π
U
Xi
→
G
i
→
1
the
extension
of
GSAFG-type
obtained
[via
α
i
]
by
considering
the
finite
étale
Galois
coverings
of
degree
a
product
of
primes
[perhaps
with
multipliticites]
∈
Σ
i
over
coverings
of
U
X
i
arising
from
Π
i
;
Π
U
Xi
Π
i
the
natural
surjection
[relative
to
α
i
].
Suppose
further
that,
for
some
l
∈
Σ
1
Σ
2
,
the
cyclotomic
characters
G
i
→
Z
×
l
have
open
image
for
i
=
1,
2.
Let
∼
φ
:
Π
1
→
Π
2
be
an
isomorphism
of
profinite
groups
such
that
φ(Δ
1
)
=
Δ
2
.
Then
there
exists
an
isomorphism
of
profinite
groups
∼
φ
U
:
Π
U
X
1
→
Π
U
X
2
that
is
compatible
with
φ,
relative
to
the
natural
surjections
Π
U
Xi
Π
i
.
More-
over,
such
an
isomorphism
is
unique
up
to
composition
with
an
inner
automor-
phism
arising
from
an
element
of
the
kernel
of
Π
U
Xi
Π
i
.
Proof.
The
construction
of
φ
U
follows
immediately
from
Corollary
3.3;
the
asserted
uniqueness
then
follows
immediately
from
our
assumption
that
the
rel-isom-DGC
holds.
Remark
3.4.1.
Just
as
in
the
case
of
Corollary
3.3
[cf.
Remark
3.3.3],
Corollary
3.4
admits
a
“tempered
version”,
when
the
base
fields
k
i
invoved
are
MLF’s.
We
leave
the
routine
details
to
the
reader.
Remark
3.4.2.
By
applying
Corollary
3.4
[cf.
also
Remarks
3.3.4,
3.4.1],
one
may
obtain
“pro-Σ
tempered”
versions
of
certain
results
of
[Mzk12]
[cf.
[Mzk12],
Theorem
1.6;
[Mzk12],
Remark
1.6.1]
concerning
the
étale
theta
function.
We
leave
the
routine
details
to
the
reader.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
71
Now
we
return
to
the
notation
introduced
at
the
beginning
of
the
present
§3:
Let
X
be
a
hyperbolic
orbicurve
over
a
field
k
of
characteristic
zero;
k
an
algebraic
closure
of
k.
Thus,
we
have
an
exact
sequence
of
fundamental
groups
1
→
π
1
(X
×
k
k)
→
π
1
(X)
→
Gal(k/k)
→
1.
Definition
3.5.
We
shall
say
that
X
is
of
strictly
Belyi
type
if
it
is
defined
over
a
number
field
and
isogenous
[cf.
§0]
to
a
hyperbolic
curve
of
genus
zero.
[Thus,
this
definition
generalizes
the
definition
of
[Mzk14],
Definition
2.3,
(i).]
Example
3.6.
Scheme-theoretic
Belyi
Cuspidalizations.
(i)
Let
P
be
a
copy
of
the
projective
line
minus
three
points
over
a
finite
Galois
extension
k
of
k;
V
an
arbitrary
hyperbolic
curve
over
k
;
U
⊆
V
a
nonempty
open
subscheme
[hence,
in
particular,
a
hyperbolic
curve
over
k
].
Suppose
that
U
[hence
also
V
]
is
defined
over
a
number
field.
Then
it
follows
from
the
existence
of
Belyi
maps
[cf.
[Belyi];
[Mzk7]]
that,
for
some
nonempty
open
subscheme
W
⊆
U
,
there
exists
a
diagram
as
follows:
W
→
⏐
⏐
β
U
→
V
P
[where
the
“→’s”
are
the
natural
open
immersions;
the
“Belyi
map”
β
is
finite
étale].
By
replacing
k
by
some
finite
extension
of
k
,
let
us
suppose
further
[for
simplicity]
that
the
cusps
of
W
are
all
defined
over
k
.
Also,
let
us
suppose
that
the
Galois
group
Gal(k/k)
is
slim.
Then,
in
the
language
of
[Mzk15],
§4,
this
situation
may
be
described
as
follows
[cf.
[Mzk15],
Definition
4.2,
(i),
where
we
take
the
extension
“1
→
Δ
→
Π
→
G
→
1”
to
be
the
extension
1
→
π
1
(P
×
k
k)
→
π
1
(P
)
→
Gal(k/k
)
→
1]:
For
some
nonnegative
integers
n,
m,
the
above
diagram
yields
a
chain
P
W
(→
P
)
(W
→)
W
n
(W
n
→)
W
n−1
.
.
.
def
(W
2
→)
W
1
=
U
(U
→)
U
m
(U
m
→)
U
m−1
.
.
.
def
(U
2
→)
U
1
=
V
whose
associated
type-chain
is
,
•,
.
.
.
,
•
[i.e.,
a
finite
étale
covering,
followed
by
n
+
m
de-cuspidalizations].
In
particular:
The
above
chain
may
thought
of
as
a
construction
of
a
“cuspidalization”
[i.e.,
result
of
passing
to
an
open
subscheme
by
removing
various
closed
points]
U
→
V
of
V
.
The
remainder
of
the
portion
of
the
theory
of
the
present
§3
concerning
Belyi
cuspidalizations
consists,
in
essence,
of
the
unraveling
of
various
consequences
of
72
SHINICHI
MOCHIZUKI
this
“chain-theoretic
formulation”
of
the
diagram
that
appears
at
the
beginning
of
the
present
item
(i).
(ii)
A
variant
of
the
discussion
of
(i)
may
be
obtained
as
follows.
In
the
notation
of
(i),
suppose
further
that
X
is
a
hyperbolic
orbicurve
of
strictly
Belyi
type
over
k,
and
that
we
have
been
given
finite
étale
coverings
V
→
X,
V
→
Q,
together
with
an
open
immersion
Q
→
P
[so
Q
is
a
hyperbolic
curve
of
genus
zero
over
k
].
Also,
[for
simplicity]
we
suppose
that
V
→
X
is
Galois,
that
U
⊆
V
descends
to
an
open
subscheme
U
X
⊆
X,
and
[by
possibly
replacing
k
by
a
finite
extension
of
k
]
that
the
cusps
of
Q
are
defined
over
k
.
Then
by
appending
to
the
chain
of
(i)
the
“finite
étale
covering”
V
→
X,
followed
by
the
“finite
étale
quotient”
V
→
Q,
followed
by
def
the
de-cuspidalizations
Q
→
Q
l
→
.
.
.
→
Q
1
=
P
[for
some
nonnegative
integer
l],
on
the
left,
and
the
“finite
étale
quotient”
V
→
X
on
the
right,
we
obtain
a
chain
def
X
V
(→
X)
(V
→)
Q
(Q
→)
Q
l
.
.
.
(Q
2
→)
Q
1
=
P
def
W
(→
P
)
(W
→)
W
n
.
.
.
(W
2
→)
W
1
=
U
(U
→)
U
m
.
.
.
def
def
(U
2
→)
U
1
=
V
(V
→)
X
∗
=
X
whose
associated
type-chain
is
,
,
•,
.
.
.
,
•,
,
•,
.
.
.
,
•,
∼
[where
the
“.
.
.
”
are
all
“•’s”],
together
with
a
terminal
isomorphism
X
∗
→
X
[i.e.,
the
identity
morphism].
In
particular,
the
above
chain
may
thought
of
as
a
construction
of
a
“cuspidalization”
U
X
→
X
of
X
via
the
construction
of
a
“cuspidalization”
U
→
V
of
V
,
equipped
with
descent
data
[i.e.,
a
suitable
collection
of
automorphisms]
with
respect
to
the
finite
étale
Galois
covering
V
→
X.
Now
by
translating
the
scheme-theoretic
discussion
of
Example
3.6
into
the
language
of
profinite
groups
via
the
theory
of
[Mzk15],
§4,
we
obtain
the
following
result.
Corollary
3.7.
(Profinite
Belyi
Cuspidalization
I:
Algorithms)
Let
D
be
a
chain-full
set
of
collections
of
partial
construction
data
[cf.
[Mzk15],
Definition
4.6,
(i)]
such
that
the
rel-isom-DGC
holds
[i.e.,
the
“relative
isomorphism
version
of
the
Grothendieck
Conjecture
for
D
holds”
—
cf.
[Mzk15],
Definition
4.6,
(ii)];
G
a
slim
profinite
group;
1
→
Δ
→
Π
→
G
→
1
an
extension
of
GSAFG-type
that
admits
partial
construction
data
(k,
X,
Σ),
where
k
is
of
characteristic
zero,
X
is
a
hyperbolic
orbicurve
of
strictly
Belyi
type
[cf.
Definition
3.5],
and
Σ
is
the
set
of
all
primes,
such
that
∼
([X],
[k],
Σ)
∈
D;
α
:
π
1
(X)
→
Π
the
corresponding
scheme-theoretic
envelope
[cf.
[Mzk15],
Definition
2.1,
(iii)],
which
is
an
isomorphism
of
profinite
groups;
∼
→
X
the
pro-finite
étale
covering
of
X
determined
by
α
[so
Π
→
Gal(
X/X)];
X
∼
k
the
resulting
algebraic
closure
of
k
[so
G
→
Gal(
k/k)].
Suppose
further
that,
for
×
some
l
∈
Σ,
the
cyclotomic
character
G
→
Z
l
has
open
image.
Thus,
by
the
theory
of
[Mzk15],
§4,
we
have
associated
categories
Chain(Π);
Chain
iso-trm
(Π);
ÉtLoc(Π)
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
73
which
may
be
constructed
via
purely
“group-theoretic”
operations
from
the
extension
of
profinite
groups
1
→
Δ
→
Π
→
G
→
1
[cf.
[Mzk15],
Definition
4.2,
(iii),
(iv),
(v);
[Mzk15],
Lemma
4.5,
(v);
the
proof
of
[Mzk15],
Theorem
4.7,
(ii)].
Then
for
every
nonempty
open
subscheme
U
X
⊆
X
defined
over
a
number
field,
the
natural
surjection
def
∼
Π
U
X
=
π
1
(U
X
)
π
1
(X)
→
Π
∼
[where
the
final
“
→
”
is
given
by
the
inverse
of
α]
—
i.e.,
“cuspidalization”
of
Π
—
may
be
constructed
via
“group-theoretic”
operations
as
follows:
(a)
For
some
normal
open
subgroup
Π
V
⊆
Π,
which
corresponds
to
a
fi-
nite
covering
V
→
X
of
hyperbolic
orbicurves,
there
exists
a
[not
neces-
sarily
unique]
Π-chain,
which
admits
an
entirely
“group-theoretic”
description,
with
associated
type-chain
,
,
•,
.
.
.
,
•,
,
•,
.
.
.
,
•,
—
cf.
Example
3.6,
(ii)
—
that
admits
a
terminal
isomorphism
with
def
the
trivial
Π-chain
[of
length
0]
such
that
if
we
write
U
=
V
×
X
U
X
,
def
Π
U
=
Π
V
×
Π
Π
U
X
,
then
the
natural
surjection
Π
U
Π
V
may
be
recovered
from
the
chain
of
“•’s”
terminating
at
the
second
to
last
group
of
the
above-mentioned
Π-chain.
(b)
The
natural
surjection
Π
U
X
Π
may
be
recovered
from
Π
U
Π
V
out
by
forming
the
“
”
[cf.
§0]
with
respect
to
the
unique
lifting
[relative
to
Π
U
Π
V
]
of
the
outer
action
of
the
finite
group
Π/Π
V
on
Π
V
to
a
group
of
outer
automorphisms
of
Π
U
.
(c)
The
decomposition
groups
of
the
closed
points
of
X
lying
in
the
complement
of
U
X
may
be
obtained
as
the
images
via
Π
U
X
Π
of
the
cuspidal
decomposition
groups
of
Π
U
X
[cf.
[Mzk15],
Lemma
4.5,
(v)].
Proof.
The
assertions
of
Corollary
3.7
follow
immediately
from
the
definitions,
together
with
the
various
references
quoted
in
the
course
of
the
“group-theoretic”
reconstruction
algorithm
described
in
the
statement
of
Corollary
3.7,
and
the
equiv-
alences
of
[Mzk15],
Theorem
4.7,
(i).
Remark
3.7.1.
Similar
remarks
to
Remarks
3.3.1,
3.3.2,
3.3.3
may
be
made
for
Corollary
3.7.
Remark
3.7.2.
In
the
situation
of
Corollary
3.7,
when
the
field
k
is
an
MLF,
one
then
obtains
an
algorithm
for
constructing
the
decomposition
groups
of
arbitrary
closed
points
of
X,
by
combining
the
algorithms
of
Corollary
3.7
—
cf.,
especially,
74
SHINICHI
MOCHIZUKI
Corollary
3.7,
(c),
which
allows
one
to
construct
the
decomposition
groups
of
those
closed
points
of
X
which
[like
U
X
!]
are
defined
over
a
number
field
—
with
the
“p-
adic
approximation
lemma”
of
[Mzk8]
[i.e.,
[Mzk8],
Lemma
3.1].
A
“Grothendieck
Conjecture-style”
version
of
this
sort
of
reconstruction
of
decomposition
groups
of
arbitrary
closed
points
of
X
may
be
found
in
[Mzk8],
Corollary
3.2.
The
“group-theoretic”
algorithm
of
Corollary
3.7
has
the
following
immediate
“Grothendieck
Conjecture-style”
consequence.
Corollary
3.8.
(Profinite
Belyi
Cuspidalization
II:
Comparison)
Let
D
be
a
chain-full
set
of
collections
of
partial
construction
data
[cf.
[Mzk15],
Definition
4.6,
(i)]
such
that
the
rel-isom-DGC
holds
[cf.
[Mzk15],
Definition
4.6,
(ii)].
For
i
=
1,
2,
let
G
i
be
a
slim
profinite
group;
1
→
Δ
i
→
Π
i
→
G
i
→
1
an
extension
of
GSAFG-type
that
admits
partial
construction
data
(k
i
,
X
i
,
Σ
i
),
where
k
i
is
of
characteristic
zero,
X
i
is
a
hyperbolic
orbicurve
of
strictly
Belyi
type
[cf.
Definition
3.5],
and
Σ
i
is
the
set
of
all
primes,
such
that
∼
([X
i
],
[k
i
],
Σ
i
)
∈
D;
α
i
:
π
1
(X
i
)
→
Π
i
the
corresponding
scheme-theoretic
en-
velope
[cf.
[Mzk15],
Definition
2.1,
(iii)],
which
is
an
isomorphism
of
profinite
i
→
X
i
the
pro-finite
étale
covering
of
X
i
determined
by
α
i
[so
groups;
X
∼
∼
i
/X
i
)];
Π
i
→
Gal(
X
k
i
the
resulting
algebraic
closure
of
k
i
[so
G
i
→
Gal(
k
i
/k
i
)].
If,
for
i
=
1,
2,
U
X
i
⊆
X
i
is
a
nonempty
open
subscheme
which
is
defined
over
a
number
field,
then
write
1
→
Δ
U
Xi
→
Π
U
Xi
→
G
i
→
1
for
the
extension
of
GSAFG-type
determined
by
[α
i
and]
the
natural
surjection
π
1
(U
X
i
)
Gal(
k
i
/k
i
)
(
∼
=
G
i
);
Π
U
Xi
Π
i
for
the
natural
surjection
[relative
to
α
i
].
Suppose
further
that,
for
some
l
∈
Σ
1
Σ
2
,
the
cyclotomic
characters
G
i
→
Z
×
l
have
open
image
for
i
=
1,
2.
Let
∼
φ
:
Π
1
→
Π
2
be
an
isomorphism
of
profinite
groups
such
that
φ(Δ
1
)
=
Δ
2
.
Then,
for
each
nonempty
open
subscheme
U
X
1
⊆
X
1
defined
over
a
number
field,
there
exist
a
nonempty
open
subscheme
U
X
2
⊆
X
2
defined
over
a
number
field
and
an
isomor-
phism
of
profinite
groups
∼
φ
U
:
Π
U
X
1
→
Π
U
X
2
that
is
compatible
with
φ,
relative
to
the
natural
surjections
Π
U
Xi
Π
i
.
More-
over,
such
an
isomorphism
φ
U
is
unique
up
to
composition
with
an
inner
auto-
morphism
arising
from
an
element
of
the
kernel
of
Π
U
Xi
Π
i
.
Proof.
The
construction
of
φ
U
for
a
suitable
nonempty
open
subscheme
U
X
2
follows
immediately
from
Corollary
3.7;
the
asserted
uniqueness
then
follows
im-
mediately
from
our
assumption
that
the
rel-isom-DGC
holds.
Remark
3.8.1.
3.8.
A
similar
remark
to
Remark
3.4.1
may
be
made
for
Corollary
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
II
75
Bibliography
[Belyi]
G.
V.
Belyi,
On
Galois
extensions
of
a
maximal
cyclotomic
field,
Izv.
Akad.
Nauk
SSSR
Ser.
Mat.
43:2
(1979),
pp.
269-276;
English
transl.
in
Math.
USSR-
Izv.
14
(1980),
pp.
247-256.
[BLR]
S.
Bosch,
W.
Lütkebohmert,
M.
Raynaud,
Néron
Models,
Ergebnisse
der
Math-
ematik
und
ihrer
Grenzgebiete
21,
Springer-Verlag
(1990).
[Ill]
L.
Illusie,
An
Overview
of
the
Work
of
K.
Fujiwara,
K.
Kato
and
C.
Nakayama
on
Logarithmic
Etale
Cohomology,
in
Cohomologies
p-adiques
et
applications
arithmétiques
(II),
P.
Berthelot,
J.-M.
Fontaine,
L.
Illusie,
K.
Kato,
M.
Rapoport,
eds.,
Asterisque
279
(2002),
pp.
271-322.
[Kato]
K.
Kato,
Toric
Singularities,
Amer.
J.
Math.
116
(1994),
pp.
1073-1099.
[Knud]
F.
F.
Knudsen,
The
Projectivity
of
the
Moduli
Space
of
Stable
Curves,
II,
Math.
Scand.
52
(1983),
pp.
161-199.
[Kobl]
N.
Koblitz,
p-adic
Numbers,
p-adic
Analysis,
and
Zeta
Functions,
Graduate
Texts
in
Mathematics
58,
Springer-Verlag
(1977).
[Mzk1]
S.
Mochizuki,
The
Geometry
of
the
Compactification
of
the
Hurwitz
Scheme,
Publ.
Res.
Inst.
Math.
Sci.
31
(1995),
pp.
355-441.
[Mzk2]
S.
Mochizuki,
Extending
Families
of
Curves
over
Log
Regular
Schemes,
J.
reine
angew.
Math.
511
(1999),
pp.
43-71.
[Mzk3]
S.
Mochizuki,
The
Local
Pro-p
Anabelian
Geometry
of
Curves,
Invent.
Math.
138
(1999),
pp.
319-423.
[Mzk4]
S.
Mochizuki,
Topics
Surrounding
the
Anabelian
Geometry
of
Hyperbolic
Curves,
Galois
Groups
and
Fundamental
Groups,
Mathematical
Sciences
Research
In-
stitute
Publications
41,
Cambridge
University
Press
(2003),
pp.
119-165.
[Mzk5]
S.
Mochizuki,
The
Absolute
Anabelian
Geometry
of
Hyperbolic
Curves,
Galois
Theory
and
Modular
Forms,
Kluwer
Academic
Publishers
(2004),
pp.
77-122.
[Mzk6]
S.
Mochizuki,
The
Absolute
Anabelian
Geometry
of
Canonical
Curves,
Kazuya
Kato’s
fiftieth
birthday,
Doc.
Math.
2003,
Extra
Vol.,
pp.
609-640.
[Mzk7]
S.
Mochizuki,
Noncritical
Belyi
Maps,
Math.
J.
Okayama
Univ.
46
(2004),
pp.
105-113.
[Mzk8]
S.
Mochizuki,
Galois
Sections
in
Absolute
Anabelian
Geometry,
Nagoya
Math.
J.
179
(2005),
pp.
17-45.
[Mzk9]
S.
Mochizuki,
Semi-graphs
of
Anabelioids,
Publ.
Res.
Inst.
Math.
Sci.
42
(2006),
pp.
221-322.
[Mzk10]
S.
Mochizuki,
Global
Solvably
Closed
Anabelian
Geometry,
Math.
J.
Okayama
Univ.
48
(2006),
pp.
57-71.
[Mzk11]
S.
Mochizuki,
Conformal
and
quasiconformal
categorical
representation
of
hy-
perbolic
Riemann
surfaces,
Hiroshima
Math.
J.
36
(2006),
pp.
405-441.
76
SHINICHI
MOCHIZUKI
[Mzk12]
S.
Mochizuki,
The
Étale
Theta
Function
and
its
Frobenioid-theoretic
Manifes-
tations,
Publ.
Res.
Inst.
Math.
Sci.
45
(2009),
pp.
227-349.
[Mzk13]
S.
Mochizuki,
A
combinatorial
version
of
the
Grothendieck
conjecture,
Tohoku
Math
J.
59
(2007),
pp.
455-479.
[Mzk14]
S.
Mochizuki,
Absolute
anabelian
cuspidalizations
of
proper
hyperbolic
curves,
J.
Math.
Kyoto
Univ.
47
(2007),
pp.
451-539.
[Mzk15]
S.
Mochizuki,
Topics
in
Absolute
Anabelian
Geometry
I:
Generalities,
J.
Math.
Sci.
Univ.
Tokyo
19
(2012),
pp.
139-242.
[Mzk16]
S.
Mochizuki,
Topics
in
Absolute
Anabelian
Geometry
III:
Global
Reconstruc-
tion
Algorithms,
RIMS
Preprint
1626
(March
2008);
updated
version
available
at
the
following
web
site:
http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html
[Mzk17]
Y.
Hoshi
and
S.
Mochizuki,
Topics
surrounding
the
combinatorial
anabelian
geometry
of
hyperbolic
curves
I:
Inertia
groups
and
profinite
Dehn
twists,
Galois-Teichmüller
Theory
and
Arithmetic
Geometry,
Adv.
Stud.
Pure
Math.
63,
Math.
Soc.
Japan
(2012),
pp.
659-811.
[MT]
S.
Mochizuki,
A.
Tamagawa,
The
algebraic
and
anabelian
geometry
of
config-
uration
spaces,
Hokkaido
Math.
J.
37
(2008),
pp.
75-131.
[SdTm]
M.
Saidi
and
A.
Tamagawa,
A
prime-to-p
version
of
Grothendieck’s
anabelian
conjecture
for
hyperbolic
curves
over
finite
fields
of
characteristic
p
>
0,
Publ.
Res.
Inst.
Math.
Sci.
45
(2009),
pp.
135-186.
[Serre]
J.-P.
Serre,
A
Course
in
Arithmetic,
Graduate
Texts
in
Mathematics
7,
Springer-
Verlag
(1973).
[Tama1]
A.
Tamagawa,
The
Grothendieck
Conjecture
for
Affine
Curves,
Compositio
Math.
109
(1997),
pp.
135-194.
[Tama2]
A.
Tamagawa,
Resolution
of
nonsingularities
of
families
of
curves,
Publ.
Res.
Inst.
Math.
Sci.
40
(2004),
pp.
1291-1336.
[Tate]
J.
Tate,
p-Divisible
Groups,
Proceedings
of
a
Conference
on
Local
Fields,
Driebergen,
Springer-Verlag
(1967),
pp.
158-183.